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Search: a002144 -id:a002144
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A002365 Numbers y such that p^2 = x^2 + y^2, 0 < x < y, p = A002144(n).
(Formerly M3430 N1391) 		+20
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
A070151 a(n) is one fourth of the even leg of the unique primitive Pythagorean triangle with hypotenuse A002144(n). +20
15
1, 3, 2, 5, 3, 10, 7, 15, 12, 20, 18, 5, 15, 28, 22, 35, 33, 13, 45, 42, 7, 15, 52, 30, 8, 65, 63, 40, 17, 78, 77, 72, 45, 68, 63, 85, 57, 10, 30, 105, 102, 70, 42, 95, 55, 110, 105, 133, 130, 12, 92, 60, 153, 152, 50, 143, 75, 138, 13, 65, 165, 27, 117, 190, 150, 187, 143, 70 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
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 x*y/2.
LINKS
T. D. Noe, Table of n, a(n) for n=1..1000
FORMULA
a(n) = A002330(n+1)*A002331(n+1)/2. - David Wasserman, May 12 2003
4*a(n) is the even positive integer with A080109(n) = A002144(n)^2 = A070079(n)^2 + (4*a(n))^2 in this unique decomposition (up to order). See A080109 for references. - 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
.................................
n = 7: a(7) = 7, A002144(7) = 53 and 53^2 = 2809 = A070079(7)^2 + (4*a(7))^2 = 45^2 + (4*7)^2 = 2025 + 784. - Wolfdieter Lang, Jan 13 2015
CROSSREFS
Cf. A002144, A002330, A002331, A070079, A080109, A144954, A144960.
KEYWORD
easy,nonn
AUTHOR
Lekraj Beedassy, May 06 2002
EXTENSIONS
Edited. New name, moved the old one to the comment section. - Wolfdieter Lang, Jan 13 2015
STATUS
approved
A054994 Numbers of the form q1^b1 * q2^b2 * q3^b3 * q4^b4 * q5^b5 * ... where q1=5, q2=13, q3=17, q4=29, q5=37, ... (A002144) and b1 >= b2 >= b3 >= b4 >= b5 >= .... +20
14
1, 5, 25, 65, 125, 325, 625, 1105, 1625, 3125, 4225, 5525, 8125, 15625, 21125, 27625, 32045, 40625, 71825, 78125, 105625, 138125, 160225, 203125, 274625, 359125, 390625, 528125, 690625, 801125, 1015625, 1185665, 1221025, 1373125, 1795625 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
This sequence is related to Pythagorean triples regarding the number of hypotenuses which are in a particular number of total Pythagorean triples and a particular number of primitive Pythagorean triples.
Least integer "mod 4 prime signature" values that are the hypotenuse of at least one primitive Pythagorean triple. - Ray Chandler, Aug 26 2004
See A097751 for definition of "mod 4 prime signature"; terms of A097752 with all prime factors of form 4*k+1.
Sequence A006339 (Least hypotenuse of n distinct Pythagorean triangles) is a subset of this sequence. - Ruediger Jehn, Jan 13 2022
LINKS
Ray Chandler, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pythagorean Triple
FORMULA
Sum_{n>=1} 1/a(n) = Product_{n>=1} 1/(1 - 1/A006278(n)) = 1.2707219403... - Amiram Eldar, Oct 20 2020
EXAMPLE
1=5^0, 5=5^1, 25=5^2, 65=5*13, 125=5^3, 325=5^2*13, 625=5^4, etc.
MATHEMATICA
maxTerm = 10^15; (* this limit gives ~ 500 terms *) maxNumberOfExponents = 9; (* this limit has to be increased until the number of reaped terms no longer changes *) bmax = Ceiling[ Log[ maxTerm]/Log[q]]; q = Reap[For[k = 0 ; cnt = 0, cnt <= maxNumberOfExponents, k++, If[PrimeQ[4*k + 1], Sow[4*k + 1]; cnt++]]][[2, 1]]; Clear[b]; b[maxNumberOfExponents + 1] = 0; iter = Sequence @@ Table[{b[k], b[k + 1], bmax[[k]]}, {k, maxNumberOfExponents, 1, -1}]; Reap[ Do[an = Product[q[[k]]^b[k], {k, 1, maxNumberOfExponents}]; If[an <= maxTerm, Print[an]; Sow[an]], Evaluate[iter]]][[2, 1]] // Flatten // Union (* Jean-François Alcover, Jan 18 2013 *)
PROG
(PARI) list(lim)=
{
my(u=[1], v=List(), w=v, pr, t=1);
forprime(p=5, ,
if(p%4>1, next);
t*=p;
if(t>lim, break);
listput(w, t)
);
for(i=1, #w,
pr=1;
for(e=1, logint(lim\=1, w[i]),
pr*=w[i];
for(j=1, #u,
t=pr*u[j];
if(t>lim, break);
listput(v, t)
)
);
if(w[i]^2<lim, u=Set(concat(Vec(v), u)); v=List());
);
Set(concat(Vec(v), u));
} \\ Charles R Greathouse IV, Dec 11 2016
(Python)
def generate_A054994():
"""generate arbitrarily many elements of the sequence.
TO_DO is a list of pairs (radius, exponents) where
"exponents" is a weakly decreasing sequence, and
radius == prod(prime_4k_plus_1(i)**j for i, j in enumerate(exponents))
An example entry is (5525, (2, 1, 1)) because 5525 = 5**2 * 13 * 17.
"""
TO_DO = {(1, ())}
while True:
radius, exponents = min(TO_DO)
yield radius #, exponents
TO_DO.remove((radius, exponents))
TO_DO.update(successors(radius, exponents))
def successors(radius, exponents):
# try to increase each exponent by 1 if possible
for i, e in enumerate(exponents):
if i==0 or exponents[i-1]>e:
# can add 1 in position i without violating monotonicity
yield (radius*prime_4k_plus_1(i), exponents[:i]+(e+1, )+exponents[i+1:])
if exponents==() or exponents[-1]>0: # add new exponent 1 at the end:
yield (radius*prime_4k_plus_1(len(exponents)), exponents+(1, ))
from sympy import isprime
primes_congruent_1_mod_4 = [5] # will be filled with 5, 13, 17, 29, 37, ...
def prime_4k_plus_1(i): # the i-th prime that is congruent to 1 mod 4
while i>=len(primes_congruent_1_mod_4): # generate primes on demand
n = primes_congruent_1_mod_4[-1]+4
while not isprime(n): n += 4
primes_congruent_1_mod_4.append(n)
return primes_congruent_1_mod_4[i]
for n, radius in enumerate(generate_A054994()):
if n==34:
print(radius)
break # print the first 35 elements
print(radius, end=", ")
# Günter Rote, Sep 12 2023
CROSSREFS
Subsequence of A097752.
Cf. A002144, A006278, A006339, A071383, A097751, A097752, A097753, A097754, A097755, A097756.
KEYWORD
easy,nonn
AUTHOR
Bernard Altschuler (Altschuler_B(AT)bls.gov), May 30 2000
EXTENSIONS
More terms from Henry Bottomley, Mar 14 2001
STATUS
approved
A070079 a(n) gives the odd leg of the unique primitive Pythagorean triangle with hypotenuse A002144(n). +20
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
A002366 Numbers x such that x^2 + y^2 = p^2 = A002144(n)^2, x < y.
(Formerly M2442 N0970) 		+20
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). +20
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
A080147 Positions of primes of the form 4*k+1 (A002144) among all primes (A000040). +20
11
3, 6, 7, 10, 12, 13, 16, 18, 21, 24, 25, 26, 29, 30, 33, 35, 37, 40, 42, 44, 45, 50, 51, 53, 55, 57, 59, 60, 62, 65, 66, 68, 70, 71, 74, 77, 78, 79, 80, 82, 84, 87, 88, 89, 97, 98, 100, 102, 104, 106, 108, 110, 112, 113, 116, 119, 121, 122, 123, 126, 127, 130, 134, 135 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
The asymptotic density of this sequence is 1/2 (by Dirichlet's theorem). - Amiram Eldar, Mar 01 2021
LINKS
Zak Seidov, Table of n, a(n) for n = 1..10000
FORMULA
A002144(n) = A000040(a(n)).
Numbers k such that prime(k) AND 2 = 0. - Gary Detlefs, Dec 26 2011
EXAMPLE
7 is in the sequence because the 7th prime, 17, is of the form 4k+1.
4 is not in the sequence because the 4th prime, 7, is not of the form 4k+1.
MAPLE
with(numtheory, ithprime); pos_of_primes_k_mod_n(300, 1, 4);
pos_of_primes_k_mod_n := proc(upto_i, k, n) local i, a; a := []; for i from 1 to upto_i do if(k = (ithprime(i) mod n)) then a := [op(a), i]; fi; od; RETURN(a); end;
with(Bits): for n from 1 to 135 do if (And(ithprime(n), 2)=0) then print(n) fi od; # Gary Detlefs, Dec 26 2011
MATHEMATICA
Select[Range[135], Mod[Prime[#], 4] == 1 &] (* Amiram Eldar, Mar 01 2021 *)
PROG
(PARI) k=0; forprime(p=2, 1e4, k++; if(p%4==1, print1(k", "))) \\ Charles R Greathouse IV, Dec 27 2011
CROSSREFS
Almost complement of A080148 (1 is excluded from both).
Cf. A000040, A002144.
KEYWORD
nonn,easy
AUTHOR
Antti Karttunen, Feb 11 2003
STATUS
approved
A334424 Decimal expansion of Product_{k>=1} (1 + 1/A002144(k)^3). +20
11
1, 0, 0, 8, 7, 6, 1, 2, 8, 4, 2, 7, 6, 0, 7, 7, 6, 3, 8, 5, 6, 5, 9, 2, 4, 1, 9, 1, 9, 6, 6, 9, 1, 7, 5, 7, 7, 9, 2, 6, 1, 9, 9, 0, 6, 6, 4, 3, 1, 7, 7, 2, 0, 6, 3, 8, 9, 2, 4, 3, 4, 7, 1, 7, 6, 1, 2, 3, 3, 6, 4, 7, 5, 9, 0, 2, 1, 4, 5, 4, 2, 4, 7, 2, 8, 4, 7, 7, 9, 2, 3, 8, 3, 9, 6, 8, 2, 9, 7, 7, 9, 1, 7, 8, 9 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
1,4
REFERENCES
B. C. Berndt, Ramanujan's notebook part IV, Springer-Verlag, 1994, p. 64-65.
LINKS
Table of n, a(n) for n=1..105.
Ph. Flajolet and I. Vardi, Zeta function expansions of some classical constants, Feb 18 1996, p. 7-8.
FORMULA
A334424 / A334425 = 105*zeta(3)/(4*Pi^3).
A334424 * A334426 = 840*zeta(3)/Pi^6.
EXAMPLE
1.008761284276077638565924191966917577926199...
CROSSREFS
Cf. A002144, A243380, A334445, A334449.
KEYWORD
nonn,cons
AUTHOR
Vaclav Kotesovec, Apr 30 2020
EXTENSIONS
a(17)-a(18) from Jinyuan Wang, Apr 30 2020
More digits from Vaclav Kotesovec, Apr 30 2020 and Jun 27 2020
STATUS
approved
A082073 First difference set of primes with 4k+1 form: A002144. +20
10
8, 4, 12, 8, 4, 12, 8, 12, 16, 8, 4, 8, 4, 24, 12, 8, 16, 8, 12, 4, 32, 4, 8, 16, 12, 8, 4, 12, 20, 4, 20, 12, 4, 20, 16, 8, 4, 8, 12, 12, 16, 8, 4, 48, 12, 20, 16, 12, 8, 16, 8, 12, 4, 24, 12, 8, 12, 4, 24, 8, 24, 24, 4, 8, 4, 24, 12, 12, 8, 24, 4, 20, 4, 48, 8, 4, 12, 24, 20, 12, 4, 8, 12 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
a(n) is divisible by 4, for all n.
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = A002144(n+1) - A002144(n).
EXAMPLE
first and second 4k+1 primes are 5 and 13, so a(1)=13-5=8;
MATHEMATICA
k=0; m=4; r=1; Do[s=Mod[Prime[n], m]; If[Equal[s, r], rp=ep; k=k+1; ep=Prime[n]; Print[(ep-rp)]; ], {n, 1, 1000}]
Differences[Select[Prime[Range[200]], Mod[#, 4]==1&]] (* Harvey P. Dale, Feb 05 2020 *)
PROG
(PARI) p=5; forprime(q=7, 1e3, if(q%4==1, print1(q-p", "); p=q)) \\ Charles R Greathouse IV, May 13 2012
CROSSREFS
Cf. A002144, A002145, A082074, A082075, A082076.
KEYWORD
nonn,easy
AUTHOR
Labos Elemer, Apr 07 2003
STATUS
approved
A334425 Decimal expansion of Product_{k>=1} (1 - 1/A002144(k)^3). +20
10
9, 9, 1, 2, 5, 1, 1, 1, 6, 2, 3, 4, 0, 9, 9, 8, 4, 4, 2, 3, 9, 7, 7, 6, 3, 6, 4, 6, 0, 9, 0, 9, 7, 7, 4, 4, 3, 3, 9, 4, 1, 5, 7, 9, 5, 0, 2, 6, 2, 9, 8, 2, 0, 0, 2, 1, 4, 1, 5, 6, 1, 0, 4, 7, 1, 7, 7, 3, 2, 7, 5, 9, 1, 4, 8, 3, 0, 0, 2, 4, 2, 1, 8, 9, 2, 0, 5, 7, 4, 1, 7, 4, 5, 0, 7, 2, 1, 7, 7, 8, 9, 7, 3, 6, 2, 0 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,1
REFERENCES
B. C. Berndt, Ramanujan's notebook part IV, Springer-Verlag, 1994, p. 64-65.
LINKS
Table of n, a(n) for n=0..105.
Ph. Flajolet and I. Vardi, Zeta function expansions of some classical constants, Feb 18 1996, p. 7-8.
R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, p. 26 (case 4 1 3 = 1/A334425).
FORMULA
A334424 / A334425 = 105*zeta(3)/(4*Pi^3).
A334425 * A334427 = 8/(7*zeta(3)).
EXAMPLE
0.991251116234099844239776364609097744339415...
CROSSREFS
Cf. A002144, A088539, A334446, A334450.
KEYWORD
nonn,cons
AUTHOR
Vaclav Kotesovec, Apr 30 2020
EXTENSIONS
More digits from Vaclav Kotesovec, Jun 27 2020
STATUS
approved
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Last modified July 16 20:46 EDT 2024. Contains 374358 sequences. (Running on oeis4.)