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A004613
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Numbers that are divisible only by primes congruent to 1 mod 4.
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39
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1, 5, 13, 17, 25, 29, 37, 41, 53, 61, 65, 73, 85, 89, 97, 101, 109, 113, 125, 137, 145, 149, 157, 169, 173, 181, 185, 193, 197, 205, 221, 229, 233, 241, 257, 265, 269, 277, 281, 289, 293, 305, 313, 317, 325, 337, 349, 353, 365, 373, 377, 389, 397, 401, 409, 421
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OFFSET
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1,2
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COMMENTS
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Also gives solutions z to x^2+y^2=z^4 with gcd(x,y,z)=1 and x,y,z positive. - John Sillcox (johnsillcox(AT)hotmail.com), Feb 20 2004
Numbers that are positive integer divisors of 1 + 4*x^2 where x is a positive integer. - Michael Somos, Jul 26 2013
Numbers k such that there is a "knight's move" of Euclidean distance sqrt(k) which allows the whole of the 2D lattice to be reached. For example, a knight which travels 4 units in any direction and then 1 unit at right angles to the first direction moves a distance sqrt(17) for each move. This knight can reach every square of an infinite chessboard.
Also 1/7 of the area of the n-th largest octagon with angles 3*Pi/4, along the perimeter of which there are only 8 nodes of the square lattice - at its vertices. - Alexander M. Domashenko, Feb 21 2024
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REFERENCES
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David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989.
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LINKS
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FORMULA
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Numbers of the form x^2 + y^2 where x is even, y is odd and gcd(x, y) = 1.
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MAPLE
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isA004613 := proc(n)
local p;
for p in numtheory[factorset](n) do
if modp(p, 4) <> 1 then
return false;
end if;
end do:
true;
end proc:
for n from 1 to 200 do
if isA004613(n) then
printf("%d, ", n) ;
end if;
# second Maple program:
q:= n-> andmap(i-> irem(i[1], 4)=1, ifactors(n)[2]):
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MATHEMATICA
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ok[1] = True; ok[n_] := And @@ (Mod[#, 4] == 1 &) /@ FactorInteger[n][[All, 1]]; Select[Range[421], ok] (* Jean-François Alcover, May 05 2011 *)
Select[Range[500], Union[Mod[#, 4]&/@(FactorInteger[#][[All, 1]])]=={1}&] (* Harvey P. Dale, Mar 08 2017 *)
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PROG
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(PARI) for(n=1, 1000, if(sumdiv(n, d, isprime(d)*if((d-1)%4, 1, 0))==0, print1(n, ", ")))
(Magma) [n: n in [1..500] | forall{d: d in PrimeDivisors(n) | d mod 4 eq 1}]; // Vincenzo Librandi, Aug 21 2012
(Haskell)
a004613 n = a004613_list !! (n-1)
a004613_list = filter (all (== 1) . map a079260 . a027748_row) [1..]
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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STATUS
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approved
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