login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Search: a070151 -id:a070151
Displaying 1-10 of 14 results found. page 1 2
     Sort: relevance | references | number | modified | created      Format: long | short | data
A002144 Pythagorean primes: primes of the form 4*k + 1.
(Formerly M3823 N1566) 		+10
482
5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 157, 173, 181, 193, 197, 229, 233, 241, 257, 269, 277, 281, 293, 313, 317, 337, 349, 353, 373, 389, 397, 401, 409, 421, 433, 449, 457, 461, 509, 521, 541, 557, 569, 577, 593, 601, 613, 617 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Rational primes that decompose in the field Q(sqrt(-1)). - N. J. A. Sloane, Dec 25 2017
These are the prime terms of A009003.
-1 is a quadratic residue mod a prime p if and only if p is in this sequence.
Sin(a(n)*Pi/2) = 1 with Pi = 3.1415..., see A070750. - Reinhard Zumkeller, May 04 2002
If at least one of the odd primes p, q belongs to the sequence, then either both or neither of the congruences x^2 = p (mod q), x^2 = q (mod p) are solvable, according to Gauss reciprocity law. - Lekraj Beedassy, Jul 17 2003
Odd primes such that binomial(p-1, (p-1)/2) == 1 (mod p). - Benoit Cloitre, Feb 07 2004
Primes that are the hypotenuse of a right triangle with integer sides. The Pythagorean triple is {A002365(n), A002366(n), a(n)}.
Also, primes of the form a^k + b^k, k > 1. - Amarnath Murthy, Nov 17 2003
The square of a(n) is the average of two other squares. This fact gives rise to a class of monic polynomials x^2 + bx + c with b = a(n) that will factor over the integers regardless of the sign of c. See A114200. - Owen Mertens (owenmertens(AT)missouristate.edu), Nov 16 2005
Also such primes p that the last digit is always 1 for the Nexus numbers of form n^p - (n-1)^p. - Alexander Adamchuk, Aug 10 2006
The set of Pythagorean primes is a proper subset of the set of positive fundamental discriminants (A003658). - Paul Muljadi, Mar 28 2008
A079260(a(n)) = 1; complement of A137409. - Reinhard Zumkeller, Oct 11 2008
From Artur Jasinski, Dec 10 2008: (Start)
If we take 4 numbers: 1, A002314(n), A152676(n), A152680(n) then multiplication table modulo a(n) is isomorphic to the Latin square:
1 2 3 4
2 4 1 3
3 1 4 2
4 3 2 1
and isomorphic to the multiplication table of {1, i, -i, -1} where i is sqrt(-1), A152680(n) is isomorphic to -1, A002314(n) with i or -i and A152676(n) vice versa -i or i. 1, A002314(n), A152676(n), A152680(n) are subfield of Galois field [a(n)]. (End)
Primes p such that the arithmetic mean of divisors of p^3 is an integer. There are 2 sequences of such primes: this one and A002145. - Ctibor O. Zizka, Oct 20 2009
Equivalently, the primes p for which the smallest extension of F_p containing the square roots of unity (necessarily F_p) contains the 4th roots of unity. In this respect, the n = 2 case of a family of sequences: see n=3 (A129805) and n=5 (A172469). - Katherine E. Stange, Feb 03 2010
Subsequence of A007969. - Reinhard Zumkeller, Jun 18 2011
A151763(a(n)) = 1.
k^k - 1 is divisible by 4*k + 1 if 4*k + 1 is a prime (see Dickson reference). - Gary Detlefs, May 22 2013
Not only are the squares of these primes the sum of two nonzero squares, but the primes themselves are also. 2 is the only prime equal to the sum of two nonzero squares and whose square is not. 2 is therefore not a Pythagorean prime. - Jean-Christophe Hervé, Nov 10 2013
The statement that these primes are the sum of two nonzero squares follows from Fermat's theorem on the sum of two squares. - Jerzy R Borysowicz, Jan 02 2019
The decompositions of the prime and its square into two nonzero squares are unique. - Jean-Christophe Hervé, Nov 11 2013. See the Dickson reference, Vol. II, (B) on p. 227. - Wolfdieter Lang, Jan 13 2015
p^e for p prime of the form 4*k+1 and e >= 1 is the sum of 2 nonzero squares. - Jon Perry, Nov 23 2014
Primes p such that the area of the isosceles triangle of sides (p, p, q) for some integer q is an integer. - Michel Lagneau, Dec 31 2014
This is the set of all primes that are the average of two squares. - Richard R. Forberg, Mar 01 2015
Numbers k such that ((k-3)!!)^2 == -1 (mod k). - Thomas Ordowski, Jul 28 2016
This is a subsequence of primes of A004431 and also of A016813. - Bernard Schott, Apr 30 2022
In addition to the comment from Jean-Christophe Hervé, Nov 10 2013: All powers as well as the products of any of these primes are the sum of two nonzero squares. They are terms of A001481, which is closed under multiplication. - Klaus Purath, Nov 19 2023
REFERENCES
David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989.
L. E. Dickson, "History of the Theory of Numbers", Chelsea Publishing Company, 1919, Vol I, page 386
L. E. Dickson, History of the Theory of Numbers, Carnegie Institution, Publ. No. 256, Vol. II, Washington D.C., 1920, p. 227.
M. du Sautoy, The Music of the Primes, Fourth Estate / HarperCollins, 2003; see p. 76.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Zak Seidov, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972.
Peter R. J. Asveld, On a Post's System of Tag. Bulletin of the EATCS 36 (1988), 96-102.
C. Banderier, Calcul de (-1/p)
J. Butcher, Mathematical Miniature 8: The Quadratic Residue Theorem, NZMS Newsletter, No. 75, April 1999.
Hing Lun Chan, Windmills of the minds: an algorithm for Fermat's Two Squares Theorem, arXiv:2112.02556 [cs.LO], 2021.
R. Chapman, Quadratic reciprocity
A. David Christopher, A partition-theoretic proof of Fermat's Two Squares Theorem, Discrete Mathematics, Volume 339, Issue 4, 6 April 2016, Pages 1410-1411.
J. E. Ewell, A Simple Proof of Fermat's Two-Square Theorem, The American Mathematical Monthly, Vol. 90, No. 9 (Nov., 1983), pp. 635-637.
Bernard Frénicle de Bessy, Traité des triangles rectangles en nombres : dans lequel plusieurs belles propriétés de ces triangles sont démontrées par de nouveaux principes, Michalet, Paris (1676) pp. 0-116; see p. 44, Consequence II.
Bernard Frénicle de Bessy, Méthode pour trouver la solution des problèmes par les exclusions. Abrégé des combinaisons. Des Quarrez magiques, in "Divers ouvrages de mathématiques et de physique, par MM. de l'Académie royale des sciences", (1693) "Troisième exemple", pp. 17-26, see in particular p. 25.
A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.
D. & C. Hazzlewood, Quadratic Reciprocity
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Lucas Lacasa, Bartolome Luque, Ignacio Gómez, and Octavio Miramontes, On a Dynamical Approach to Some Prime Number Sequences, Entropy 20.2 (2018): 131, also arXiv:1802.08349 [math.NT], 2018.
R. C. Laubenbacher and D. J. Pengelley, Eisenstein's Misunderstood Geometric Proof of the Quadratic Reciprocity Theorem
R. C. Laubenbacher and D. J. Pengelley, Gauss, Eisenstein and the 'third' proof of the Quadratic Reciprocity Theorem
K. Matthews, Serret's algorithm based Server
Carlos Rivera, Puzzle 968. Another property of primes 4m+1, The Prime Puzzles & Problems Connection.
D. Shanks, Review of "K. E. Kloss et al., Class number of primes of the form 4n+1", Math. Comp., 23 (1969), 213-214. [Annotated scanned preprint of review]
S. A. Shirali, A family portrait of primes-a case study in discrimination, Math. Mag. Vol. 70, No. 4 (Oct., 1997), pp. 263-272.
Rosemary Sullivan and Neil Watling, Independent divisibility pairs on the set of integers from 1 to n, INTEGERS 13 (2013) #A65.
Eric Weisstein's World of Mathematics, Wilson's Theorem
Eric Weisstein's World of Mathematics, Pythagorean Triples
Wikipedia, Quadratic reciprocity
Wolfram Research, The Gauss Reciprocity Law
G. Xiao, Two squares
D. Zagier, A One-Sentence Proof That Every Prime p == 1 (mod 4) Is a Sum of Two Squares, Am. Math. Monthly, Vol. 97, No. 2 (Feb 1990), p. 144. [From Wolfdieter Lang, Jan 17 2015 (thanks to Charles Nash)]
Index to sequences related to decomposition of primes in quadratic fields
FORMULA
Odd primes of form x^2 + y^2, (x=A002331, y=A002330, with x < y) or of form u^2 + 4*v^2, (u = A002972, v = A002973, with u odd). - Lekraj Beedassy, Jul 16 2004
p^2 - 1 = 12*Sum_{i = 0..floor(p/4)} floor(sqrt(i*p)) where p = a(n) = 4*n + 1. [Shirali]
a(n) = A000290(A002972(n)) + A000290(2*A002973(n)) = A000290(A002331(n+1)) + A000290(A002330(n+1)). - Reinhard Zumkeller, Feb 16 2010
a(n) = A002972(n)^2 + (2*A002973(n))^2, n >= 1. See the Jean-Christophe Hervé Nov 11 2013 comment. - Wolfdieter Lang, Jan 13 2015
a(n) = 4*A005098(n) + 1. - Zak Seidov, Sep 16 2018
From Vaclav Kotesovec, Apr 30 2020: (Start)
Product_{k>=1} (1 - 1/a(k)^2) = A088539.
Product_{k>=1} (1 + 1/a(k)^2) = A243380.
Product_{k>=1} (1 - 1/a(k)^3) = A334425.
Product_{k>=1} (1 + 1/a(k)^3) = A334424.
Product_{k>=1} (1 - 1/a(k)^4) = A334446.
Product_{k>=1} (1 + 1/a(k)^4) = A334445.
Product_{k>=1} (1 - 1/a(k)^5) = A334450.
Product_{k>=1} (1 + 1/a(k)^5) = A334449. (End)
From Vaclav Kotesovec, May 05 2020: (Start)
Product_{k>=1} (1 + 1/A002145(k)) / (1 + 1/a(k)) = Pi/(4*A064533^2) = 1.3447728438248695625516649942427635670667319092323632111110962...
Product_{k>=1} (1 - 1/A002145(k)) / (1 - 1/a(k)) = Pi/(8*A064533^2) = 0.6723864219124347812758324971213817835333659546161816055555481... (End)
Sum_{k >= 1} 1/a(k)^s = (1/2) * Sum_{n >= 1 odd numbers} moebius(n) * log((2*n*s)! * zeta(n*s) * abs(EulerE(n*s - 1)) / (Pi^(n*s) * 2^(2*n*s) * BernoulliB(2*n*s) * (2^(n*s) + 1) * (n*s - 1)!))/n, s >= 3 odd number. - Dimitris Valianatos, May 21 2020
Legendre symbol (-1, a(n)) = +1, for n >= 1. - Wolfdieter Lang, Mar 03 2021
EXAMPLE
The following table shows the relationship between several closely related sequences:
Here p = A002144 = primes == 1 (mod 4), p = a^2+b^2 with a < b;
a = A002331, b = A002330, t_1 = ab/2 = A070151;
p^2 = c^2 + d^2 with c < d; c = A002366, d = A002365,
t_2 = 2ab = A145046, t_3 = b^2 - a^2 = A070079,
with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2-a^2).
---------------------------------
p a b t_1 c d t_2 t_3 t_4
---------------------------------
5 1 2 1 3 4 4 3 6
13 2 3 3 5 12 12 5 30
17 1 4 2 8 15 8 15 60
29 2 5 5 20 21 20 21 210
37 1 6 3 12 35 12 35 210
41 4 5 10 9 40 40 9 180
53 2 7 7 28 45 28 45 630
...
a(7) = 53 = A002972(7)^2 + (2*A002973(7))^2 = 7^2 + (2*1)^2 = 49 + 4, and this is the only way. - Wolfdieter Lang, Jan 13 2015
MAPLE
a := []; for n from 1 to 500 do if isprime(4*n+1) then a := [op(a), 4*n+1]; fi; od: A002144 := n->a[n];
# alternative
A002144 := proc(n)
option remember ;
local a;
if n = 1 then
5;
else
for a from procname(n-1)+4 by 4 do
if isprime(a) then
return a;
end if;
end do:
end if;
end proc:
seq(A002144(n), n=1..100) ; # R. J. Mathar, Jan 31 2024
MATHEMATICA
Select[4*Range[140] + 1, PrimeQ[ # ] &] (* Stefan Steinerberger, Apr 16 2006 *)
Select[Prime[Range[150]], Mod[#, 4]==1&] (* Harvey P. Dale, Jan 28 2021 *)
PROG
(Haskell)
a002144 n = a002144_list !! (n-1)
a002144_list = filter ((== 1) . a010051) [1, 5..]
-- Reinhard Zumkeller, Mar 06 2012, Feb 22 2011
(Magma) [a: n in [0..200] | IsPrime(a) where a is 4*n + 1 ]; // Vincenzo Librandi, Nov 23 2014
(PARI) select(p->p%4==1, primes(1000))
(PARI)
A002144_next(p=A2144[#A2144])={until(isprime(p+=4), ); p} /* NB: p must be of the form 4k+1. Beyond primelimit, this is *much* faster than forprime(p=..., , p%4==1 && return(p)). */
A2144=List(5); A002144(n)={while(#A2144<n, listput(A2144, A002144_next())); A2144[n]}
\\ M. F. Hasler, Jul 06 2024
(Python)
from sympy import prime
A002144 = [n for n in (prime(x) for x in range(1, 10**3)) if not (n-1) % 4]
# Chai Wah Wu, Sep 01 2014
(Python)
from sympy import isprime
print(list(filter(isprime, range(1, 618, 4)))) # Michael S. Branicky, May 13 2021
(SageMath)
def A002144_list(n): # returns all Pythagorean primes <= n
return [x for x in prime_range(5, n+1) if x % 4 == 1]
A002144_list(617) # Peter Luschny, Sep 12 2012
CROSSREFS
Cf. A002145, A002314, A002476, A002972, A002973, A003658, A004431, A007519, A010051, A016813, A076339, A094407.
Cf. A114200, A133870, A142925, A152676, A152680, A173330, A173331, A208177, A208178, A334912.
Cf. A004613 (multiplicative closure).
Apart from initial term, same as A002313.
For values of n see A005098.
Primes in A020668.
KEYWORD
nonn,easy,nice
AUTHOR
N. J. A. Sloane
STATUS
approved
A002330 Value of y in the solution to p = x^2 + y^2, x <= y, with prime p = A002313(n).
(Formerly M0462 N0169) 		+10
39
1, 2, 3, 4, 5, 6, 5, 7, 6, 8, 8, 9, 10, 10, 8, 11, 10, 11, 13, 10, 12, 14, 15, 13, 15, 16, 13, 14, 16, 17, 13, 14, 16, 18, 17, 18, 17, 19, 20, 20, 15, 17, 20, 21, 19, 22, 20, 21, 19, 20, 24, 23, 24, 18, 19, 25, 22, 25, 23, 26, 26, 22, 27, 26, 20, 25, 22, 26, 28, 25 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
REFERENCES
A. J. C. Cunningham, Quadratic Partitions. Hodgson, London, 1904, p. 1.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
John Brillhart, Note on representing a prime as a sum of two squares, Math. Comp. 26 (1972), pp. 1011-1013.
A. J. C. Cunningham, Quadratic Partitions, Hodgson, London, 1904. [Annotated scans of selected pages]
K. Matthews, Serret's algorithm Server.
J. Todd, A problem on arc tangent relations, Amer. Math. Monthly, 56 (1949), 517-528.
Eric Weisstein's World of Mathematics, Fermat's 4n Plus 1 Theorem.
FORMULA
a(n) = A096029(n) + A096030(n) + 1, for n>1. - Lekraj Beedassy, Jul 21 2004
a(n+1) = Max(A002972(n), 2*A002973(n)). - Reinhard Zumkeller, Feb 16 2010
EXAMPLE
The following table shows the relationship between several closely related sequences:
Here p = A002144 = primes == 1 (mod 4), p = a^2+b^2 with a < b;
a = A002331, b = A002330, t_1 = ab/2 = A070151;
p^2 = c^2+d^2 with c < d; c = A002366, d = A002365,
t_2 = 2ab = A145046, t_3 = b^2-a^2 = A070079,
with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2-a^2).
---------------------------------
.p..a..b..t_1..c...d.t_2.t_3..t_4
---------------------------------
.5..1..2...1...3...4...4...3....6
13..2..3...3...5..12..12...5...30
17..1..4...2...8..15...8..15...60
29..2..5...5..20..21..20..21..210
37..1..6...3..12..35..12..35..210
41..4..5..10...9..40..40...9..180
53..2..7...7..28..45..28..45..630
.................................
MAPLE
a := []; for x from 0 to 50 do for y from x to 50 do p := x^2+y^2; if isprime(p) then a := [op(a), [p, x, y]]; fi; od: od: writeto(trans); for i from 1 to 158 do lprint(a[i]); od: # then sort the triples in "trans"
MATHEMATICA
Flatten[#, 1]&[Table[PowersRepresentations[Prime[k], 2, 2], {k, 1, 142}]][[All, 2]] (* Jean-François Alcover, Jul 05 2011 *)
PROG
(PARI) f(p)=my(s=lift(sqrt(Mod(-1, p))), x=p, t); if(s>p/2, s=p-s); while(s^2>p, t=s; s=x%s; x=t); s
forprime(p=2, 1e3, if(p%4-3, print1(f(p)", "))) \\ Charles R Greathouse IV, Apr 24 2012
(PARI) do(p)=qfbsolve(Qfb(1, 0, 1), p)[1]
forprime(p=2, 1e3, if(p%4-3, print1(do(p)", "))) \\ Charles R Greathouse IV, Sep 26 2013
(PARI) print1(1); forprimestep(p=5, 1e3, 4, print1(", "qfbcornacchia(1, p)[1])) \\ Charles R Greathouse IV, Sep 15 2021
CROSSREFS
Cf. A002331, A002313, A002144.
KEYWORD
nonn
AUTHOR
N. J. A. Sloane
STATUS
approved
A002331 Values of x in the solution to p = x^2 + y^2, x <= y, with prime p = A002313(n).
(Formerly M0096 N0033) 		+10
23
1, 1, 2, 1, 2, 1, 4, 2, 5, 3, 5, 4, 1, 3, 7, 4, 7, 6, 2, 9, 7, 1, 2, 8, 4, 1, 10, 9, 5, 2, 12, 11, 9, 5, 8, 7, 10, 6, 1, 3, 14, 12, 7, 4, 10, 5, 11, 10, 14, 13, 1, 8, 5, 17, 16, 4, 13, 6, 12, 1, 5, 15, 2, 9, 19, 12, 17, 11, 5, 14, 10, 18, 4, 6, 16, 20, 19, 10, 13, 4, 6, 15, 22, 11, 3, 5 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
REFERENCES
A. J. C. Cunningham, Quadratic Partitions. Hodgson, London, 1904, p. 1.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
A. T. Benjamin and D. Zeilberger, Pythagorean primes and palindromic continued fractionsINTEGERS 5(1) (2005) #A30
John Brillhart, Note on representing a prime as a sum of two squares, Math. Comp. 26 (1972), pp. 1011-1013.
A. J. C. Cunningham, Quadratic Partitions, Hodgson, London, 1904. [Annotated scans of selected pages]
K. Matthews, Serret's algorithm Server.
J. Todd, A problem on arc tangent relations, Amer. Math. Monthly, 56 (1949), 517-528.
Eric Weisstein's World of Mathematics, Fermat's 4n Plus 1 Theorem.
FORMULA
a(n) = A096029(n) - A096030(n) for n > 1. - Lekraj Beedassy, Jul 16 2004
a(n+1) = Min(A002972(n), 2*A002973(n)). - Reinhard Zumkeller, Feb 16 2010
a(n) = A363051(A002313(n)). - R. J. Mathar, Jan 31 2024
EXAMPLE
The following table shows the relationship
between several closely related sequences:
Here p = A002144 = primes == 1 (mod 4), p = a^2+b^2 with a < b;
a = A002331, b = A002330, t_1 = ab/2 = A070151;
p^2 = c^2+d^2 with c < d; c = A002366, d = A002365,
t_2 = 2ab = A145046, t_3 = b^2-a^2 = A070079,
with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2-a^2).
---------------------------------
.p..a..b..t_1..c...d.t_2.t_3..t_4
---------------------------------
.5..1..2...1...3...4...4...3....6
13..2..3...3...5..12..12...5...30
17..1..4...2...8..15...8..15...60
29..2..5...5..20..21..20..21..210
37..1..6...3..12..35..12..35..210
41..4..5..10...9..40..40...9..180
53..2..7...7..28..45..28..45..630
.................................
MAPLE
See A002330 for Maple program.
# alternative
A002331 := proc(n)
A363051(A002313(n)) ;
end proc:
seq(A002331(n), n=1..100) ; # R. J. Mathar, Feb 01 2024
MATHEMATICA
pmax = 1000; x[p_] := Module[{x, y}, x /. ToRules[Reduce[0 <= x <= y && x^2 + y^2 == p, {x, y}, Integers]]]; For[n=1; p=2, p<pmax, p = NextPrime[p], If[Mod[p, 4] == 1 || Mod[p, 4] == 2, a[n] = x[p]; Print["a(", n, ") = ", a[n]]; n++]]; Array[a, n-1] (* Jean-François Alcover, Feb 26 2016 *)
PROG
(PARI) f(p)=my(s=lift(sqrt(Mod(-1, p))), x=p, t); if(s>p/2, s=p-s); while(s^2>p, t=s; s=x%s; x=t); s
forprime(p=2, 1e3, if(p%4-3, print1(sqrtint(p-f(p)^2)", ")))
\\ Charles R Greathouse IV, Apr 24 2012
(PARI) do(p)=qfbsolve(Qfb(1, 0, 1), p)[2]
forprime(p=2, 1e3, if(p%4-3, print1(do(p)", "))) \\ Charles R Greathouse IV, Sep 26 2013
CROSSREFS
Cf. A002330, A002313, A002144, A027862 (locates y=x+1).
KEYWORD
nonn
AUTHOR
N. J. A. Sloane
STATUS
approved
A002365 Numbers y such that p^2 = x^2 + y^2, 0 < x < y, p = A002144(n).
(Formerly M3430 N1391) 		+10
15
4, 12, 15, 21, 35, 40, 45, 60, 55, 80, 72, 99, 91, 112, 105, 140, 132, 165, 180, 168, 195, 221, 208, 209, 255, 260, 252, 231, 285, 312, 308, 288, 299, 272, 275, 340, 325, 399, 391, 420, 408, 351, 425, 380, 459, 440, 420, 532, 520, 575, 465, 551, 612, 608, 609 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
REFERENCES
A. J. C. Cunningham, Quadratic and Linear Tables. Hodgson, London, 1927, pp. 77-79.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 60.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
T. D. Noe, Table of n, a(n) for n = 1..1000
A. J. C. Cunningham, Quadratic and Linear Tables, Hodgson, London, 1927 [Annotated scanned copy of selected pages]
EXAMPLE
The following table shows the relationship
between several closely related sequences:
Here p = A002144 = primes == 1 mod 4, p = a^2+b^2 with a < b;
a = A002331, b = A002330, t_1 = ab/2 = A070151;
p^2 = c^2+d^2 with c < d; c = A002366, d = A002365,
t_2 = 2ab = A145046, t_3 = b^2-a^2 = A070079,
with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2-a^2).
---------------------------------
.p..a..b..t_1..c...d.t_2.t_3..t_4
---------------------------------
.5..1..2...1...3...4...4...3....6
13..2..3...3...5..12..12...5...30
17..1..4...2...8..15...8..15...60
29..2..5...5..20..21..20..21..210
37..1..6...3..12..35..12..35..210
41..4..5..10...9..40..40...9..180
53..2..7...7..28..45..28..45..630
.................................
3^2 + 4^2 = 5^2, giving x=3, y=4, p=5 and we have the first terms of A002366, the present sequence and A002144.
CROSSREFS
Cf. A002144, A002366.
KEYWORD
nonn
AUTHOR
N. J. A. Sloane
EXTENSIONS
More terms from Ray Chandler, Jun 23 2004
Revised definition from M. F. Hasler, Feb 24 2009
STATUS
approved
A070079 a(n) gives the odd leg of the unique primitive Pythagorean triangle with hypotenuse A002144(n). +10
14
3, 5, 15, 21, 35, 9, 45, 11, 55, 39, 65, 99, 91, 15, 105, 51, 85, 165, 19, 95, 195, 221, 105, 209, 255, 69, 115, 231, 285, 25, 75, 175, 299, 225, 275, 189, 325, 399, 391, 29, 145, 351, 425, 261, 459, 279, 341, 165, 231, 575, 465, 551, 35, 105, 609, 315, 589, 385, 675 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Consider sequence A002144 of primes congruent to 1 (mod 4) and equal to x^2 + y^2, with y>x given by A002330 and A002331; sequence gives values y^2 - x^2.
Odd legs of primitive Pythagorean triangles with unique (prime) hypotenuse (A002144), sorted on the latter. Corresponding even legs are given by 4*A070151 (or A145046). - Lekraj Beedassy, Jul 22 2005
LINKS
T. D. Noe, Table of n, a(n) for n=1..1000
FORMULA
a(n)=A079886(n)*A079887(n) - Benoit Cloitre, Jan 13 2003
a(n) is the odd positive integer with A080109(n) = A002144(n)^2 = a(n)^2 + (4*A070151(n))^2, in this unique decomposition into positive squares (up to order). See the Lekraj Beedassy, comment. - Wolfdieter Lang, Jan 13 2015
EXAMPLE
The following table shows the relationship
between several closely related sequences:
Here p = A002144 = primes == 1 mod 4, p = a^2+b^2 with a < b;
a = A002331, b = A002330, t_1 = ab/2 = A070151;
p^2 = c^2+d^2 with c < d; c = A002366, d = A002365,
t_2 = 2ab = A145046, t_3 = b^2-a^2 = A070079,
with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2-a^2).
---------------------------------
.p..a..b..t_1..c...d.t_2.t_3..t_4
---------------------------------
.5..1..2...1...3...4...4...3....6
13..2..3...3...5..12..12...5...30
17..1..4...2...8..15...8..15...60
29..2..5...5..20..21..20..21..210
37..1..6...3..12..35..12..35..210
41..4..5..10...9..40..40...9..180
53..2..7...7..28..45..28..45..630
.................................
MATHEMATICA
pp = Select[ Range[200] // Prime, Mod[#, 4] == 1 &]; f[p_] := y^2 - x^2 /. ToRules[ Reduce[0 <= x <= y && p == x^2 + y^2, {x, y}, Integers]]; A070079 = f /@ pp (* Jean-François Alcover, Jan 15 2015 *)
CROSSREFS
Cf. A002144, A002330, A002331, A080109, A070151
KEYWORD
easy,nonn
AUTHOR
Lekraj Beedassy, May 06 2002
EXTENSIONS
More terms from Benoit Cloitre, Jan 13 2003
Edited: Used a different name and moved old name to the comment section. - Wolfdieter Lang, Jan 13 2015
STATUS
approved
A145046 a(n) = 2*A002330(n+1) * A002331(n+1). +10
14
4, 12, 8, 20, 12, 40, 28, 60, 48, 80, 72, 20, 60, 112, 88, 140, 132, 52, 180, 168, 28, 60, 208, 120, 32, 260, 252, 160, 68, 312, 308, 288, 180, 272, 252, 340, 228, 40, 120, 420, 408, 280, 168, 380, 220, 440, 420, 532, 520, 48, 368, 240, 612, 608, 200, 572, 300, 552, 52, 260 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
LINKS
Table of n, a(n) for n=0..59.
EXAMPLE
The following table shows the relationship
between several closely related sequences:
Here p = A002144 = primes == 1 mod 4, p = a^2+b^2 with a < b;
a = A002331, b = A002330, t_1 = ab/2 = A070151;
p^2 = c^2+d^2 with c < d; c = A002366, d = A002365,
t_2 = 2ab = A145046, t_3 = b^2-a^2 = A070079,
with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2-a^2).
---------------------------------
.p..a..b..t_1..c...d.t_2.t_3..t_4
---------------------------------
.5..1..2...1...3...4...4...3....6
13..2..3...3...5..12..12...5...30
17..1..4...2...8..15...8..15...60
29..2..5...5..20..21..20..21..210
37..1..6...3..12..35..12..35..210
41..4..5..10...9..40..40...9..180
53..2..7...7..28..45..28..45..630
.................................
CROSSREFS
4 times A070151, or (apart from initial term) twice A145019.
Cf. A070079.
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Feb 25 2009
STATUS
approved
A002366 Numbers x such that x^2 + y^2 = p^2 = A002144(n)^2, x < y.
(Formerly M2442 N0970) 		+10
13
3, 5, 8, 20, 12, 9, 28, 11, 48, 39, 65, 20, 60, 15, 88, 51, 85, 52, 19, 95, 28, 60, 105, 120, 32, 69, 115, 160, 68, 25, 75, 175, 180, 225, 252, 189, 228, 40, 120, 29, 145, 280, 168, 261, 220, 279, 341, 165, 231, 48, 368, 240, 35, 105, 200, 315, 300, 385, 52, 260, 259 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
REFERENCES
A. J. C. Cunningham, Quadratic and Linear Tables. Hodgson, London, 1927, pp. 77-79.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 60.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
T. D. Noe, Table of n, a(n) for n = 1..1000
A. J. C. Cunningham, Quadratic and Linear Tables, Hodgson, London, 1927 [Annotated scanned copy of selected pages]
EXAMPLE
The following table shows the relationship
between several closely related sequences:
Here p = A002144 = primes == 1 mod 4, p = a^2+b^2 with a < b;
a = A002331, b = A002330, t_1 = ab/2 = A070151;
p^2 = c^2+d^2 with c < d; c = A002366, d = A002365,
t_2 = 2ab = A145046, t_3 = b^2-a^2 = A070079,
with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2-a^2).
---------------------------------
.p..a..b..t_1..c...d.t_2.t_3..t_4
---------------------------------
.5..1..2...1...3...4...4...3....6
13..2..3...3...5..12..12...5...30
17..1..4...2...8..15...8..15...60
29..2..5...5..20..21..20..21..210
37..1..6...3..12..35..12..35..210
41..4..5..10...9..40..40...9..180
53..2..7...7..28..45..28..45..630
.................................
CROSSREFS
Cf. A002313, A002330, A002331.
KEYWORD
nonn
AUTHOR
N. J. A. Sloane
EXTENSIONS
More terms from Ray Chandler, Jun 23 2004
Corrected definition to require p=A002144(n), which defines the order of the terms. - M. F. Hasler, Feb 24 2009
STATUS
approved
A080109 Square of primes of the form 4k+1 (A002144). +10
11
25, 169, 289, 841, 1369, 1681, 2809, 3721, 5329, 7921, 9409, 10201, 11881, 12769, 18769, 22201, 24649, 29929, 32761, 37249, 38809, 52441, 54289, 58081, 66049, 72361, 76729, 78961, 85849, 97969, 100489, 113569, 121801, 124609, 139129 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
a(n) is the sum of two positive squares in only one way. See the Dickson reference, (B) p. 227.
a(n) is the hypotenuse of two and only two right triangles with integral legs (modulo leg exchange). See the Dickson reference, (A) p. 227.
In 1640 Fermat generalized the 3,4,5 triangle with the theorem: A prime of the form 4n+1 is the hypotenuse of one and only one right triangle with integral arms. The square of a prime of the form 4n+1 is the hypotenuse of two and only two... The cube of three and only three...
REFERENCES
L. E. Dickson, History of the Theory of Numbers, Volume II, Diophantine Analysis. Carnegie Institution Publ. No. 256, Vol II, Washington, DC, 1920, p. 227.
Morris Kline, Mathematical Thought from Ancient to Modern Times, 1972, pp. 275-276.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = A002144(n)^2 = A070079(n)^2 + (4*A070151(n))^2, for n >= 1. - Wolfdieter Lang, Jan 13 2015
From Amiram Eldar, Dec 02 2022: (Start)
Product_{n>=1} (1 + 1/a(n)) = A243380
Product_{n>=1} (1 - 1/a(n)) = A088539. (End)
EXAMPLE
a(7) = 2809 is the hypotenuse of triangles 1241, 2520, 2809 and 1484, 2385, 2809, and only of these.
a(7) = 53^2 = 2809 = 45^2 + (4*7)^2, and this is the only way. - Wolfdieter Lang, Jan 13 2015
MATHEMATICA
Select[4 Range[96] + 1, PrimeQ]^2 (* Michael De Vlieger, Dec 27 2016 *)
PROG
(PARI) fermat(n) = { for(x=1, n, y=4*x+1; if(isprime(y), print1(y^2" ")) ) }
CROSSREFS
Cf. A002144, A080175. - Wolfdieter Lang, Jan 13 2015
Cf. A070079, A070151, A088539, A243380.
KEYWORD
nonn,easy
AUTHOR
Cino Hilliard, Mar 16 2003
EXTENSIONS
Edited: Name changed, part of old name as comment. Comments added and changed. Dickson reference added. - Wolfdieter Lang, Jan 13 2015
STATUS
approved
A277557 The ordered image of the 1-to-1 mapping of an integer ordered pair (x,y) into an integer using Cantor's pairing function, where 0 < x < y, gcd(x,y)=1 and x+y odd. +10
4
8, 18, 19, 32, 33, 34, 50, 52, 53, 72, 73, 74, 75, 76, 98, 99, 100, 101, 102, 103, 128, 131, 133, 134, 162, 163, 164, 165, 166, 167, 168, 169, 200, 201, 202, 203, 204, 205, 206, 207, 208, 242, 244, 247, 248, 250, 251, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 338 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
The mapping of the ordered pair (x,y) to an integer uses Cantor's pairing function to generate the integer as (x+y)(x+y+1)/2+y. Also for every ordered pair (x,y) such that 0 < x < y, gcd(x,y)=1 and x+y odd, there exists a primitive Pythagorean triple (PPT) (a, b, c) such that a = y^2-x^2, b = 2xy, c = x^2+y^2. Therefore each term in the sequence represents a unique PPT.
Numbers n for which 0 < A025581(n) < A002262(n) and A025581(n)+A002262(n) is odd, and gcd(A025581(n), A002262(n)) = 1. [The definition expressed with A-numbers.] - Antti Karttunen, Nov 02 2016
See also the triangle T(y, x) with the values for PPTs given in A278147. - Wolfdieter Lang, Nov 24 2016
LINKS
Table of n, a(n) for n=1..59.
Wikipedia, Cantor's pairing function, and Pythagorean triple
EXAMPLE
a(5)=33 because the ordered pair (2,5) maps to 33 by Cantor's pairing function (see below) and is the 5th such occurrence. Also x=2, y=5 generates a PPT with sides (21,20,29).
Note: Cantor's pairing function is simply A001477 in its two-argument tabular form A001477(k, n) = n + (k+n)*(k+n+1)/2, thus A001477(2,5) = 5 + (2+5)*(2+5+1)/2 = 33. - Antti Karttunen, Nov 02 2016
MATHEMATICA
Cantor[{i_, j_}] := (i+j)(i+j+1)/2+j; getparts[n_] := Reverse@Select[Reverse[IntegerPartitions[n, {2}], 2], GCD@@#==1 &]; pairs=Flatten[Table[getparts[2n+1], {n, 1, 20}], 1]; Table[Cantor[pairs[[n]]], {n, 1, Length[pairs]}]
PROG
(Scheme, with Antti Karttunen's IntSeq-library)
(define A277557 (MATCHING-POS 1 1 (lambda (n) (let ((x (A025581 n)) (y (A002262 n))) (and (not (zero? x)) (< x y) (odd? (+ x y)) (= 1 (gcd x y))))))) ;; Antti Karttunen, Nov 02 2016
CROSSREFS
Cf. A001477, A002262, A025581, A243808, A277632.
Cf. A020882 (is obtained when A048147(a(n)) is sorted into ascending order), A008846 (same with duplicates removed).
Cf. A020887, A020888, A120427, A024362, A024406, A046079, A046087, A070151, A156678, A156679, A156680, A156683, A156685, A222946, A278147.
KEYWORD
nonn
AUTHOR
Frank M Jackson, Oct 19 2016
STATUS
approved
A145010 a(n) = area of Pythagorean triangle with hypotenuse p, where p = A002144(n) = n-th prime == 1 (mod 4). +10
3
6, 30, 60, 210, 210, 180, 630, 330, 1320, 1560, 2340, 990, 2730, 840, 4620, 3570, 5610, 4290, 1710, 7980, 2730, 6630, 10920, 12540, 4080, 8970, 14490, 18480, 9690, 3900, 11550, 25200, 26910, 30600, 34650, 32130, 37050, 7980, 23460, 6090, 29580, 49140, 35700 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Pythagorean primes, i.e., primes of the form p = 4k+1 = A002144(n), have exactly one representation as sum of two squares: A002144(n) = x^2+y^2 = A002330(n+1)^2+A002331(n+1)^2. The corresponding (primitive) integer-sided right triangle with sides { 2xy, |x^2-y^2| } = { A002365(n), A002366(n) } has area xy|x^2-y^2| = a(n). For n>1 this is a(n) = 30*A068386(n).
LINKS
T. D. Noe, Table of n, a(n) for n = 1..1000
FORMULA
a(n) = A002365(n)*A002366(n)/2.
a(n) = x*y*(x^2-y^2), where x = A002330(n+1), y = A002331(n+1).
EXAMPLE
The following table shows the relationship between several closely related sequences:
Here p = A002144 = primes == 1 (mod 4), p = a^2+b^2 with a < b;
a = A002331, b = A002330, t_1 = ab/2 = A070151;
p^2 = c^2+d^2 with c < d; c = A002366, d = A002365,
t_2 = 2ab = A145046, t_3 = b^2-a^2 = A070079,
with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2-a^2).
---------------------------------
p a b t_1 c d t_2 t_3 t_4
---------------------------------
5 1 2 1 3 4 4 3 6
13 2 3 3 5 12 12 5 30
17 1 4 2 8 15 8 15 60
29 2 5 5 20 21 20 21 210
37 1 6 3 12 35 12 35 210
41 4 5 10 9 40 40 9 180
53 2 7 7 28 45 28 45 630
MATHEMATICA
Reap[For[p = 2, p < 500, p = NextPrime[p], If[Mod[p, 4] == 1, area = x*y/2 /. ToRules[Reduce[0 < x <= y && p^2 == x^2 + y^2, {x, y}, Integers]]; Sow[area]]]][[2, 1]] (* Jean-François Alcover, Feb 04 2015 *)
PROG
(PARI) forprime(p=1, 499, p%4==1 | next; t=[p, lift(-sqrt(Mod(-1, p)))]; while(t[1]^2>p, t=[t[2], t[1]%t[2]]); print1(t[1]*t[2]*(t[1]^2-t[2]^2)", "))
(PARI) {Q=Qfb(1, 0, 1); forprime(p=1, 499, p%4==1|next; t=qfbsolve(Q, p); print1(t[1]*t[2]*(t[1]^2-t[2]^2)", "))} \\ David Broadhurst
CROSSREFS
Cf. A002144, A002365, A002366, A144954.
KEYWORD
nonn
AUTHOR
M. F. Hasler, Feb 24 2009
STATUS
approved
page 1 2

Search completed in 0.015 seconds

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 26 13:33 EDT 2024. Contains 375456 sequences. (Running on oeis4.)