cp's OEIS Frontend

A057653 Odd numbers of form x^2 + y^2.

%I A057653 #37 Jul 02 2025 16:02:00
%S A057653 1,5,9,13,17,25,29,37,41,45,49,53,61,65,73,81,85,89,97,101,109,113,
%T A057653 117,121,125,137,145,149,153,157,169,173,181,185,193,197,205,221,225,
%U A057653 229,233,241,245,257,261,265,269,277,281,289,293,305,313,317,325,333,337
%N A057653 Odd numbers of form x^2 + y^2.
%C A057653 Numbers with only odd prime factors and such that all prime factors congruent to 3 modulo 4 occur to an even exponent. - _Jean-Christophe Hervé_, Oct 24 2015
%C A057653 Odd terms of A020668. - _Altug Alkan_, Nov 19 2015
%C A057653 Also one half of the numbers that are the sum of two odd squares (without multiplicity). See A097269 for twice the numbers. - _Wolfdieter Lang_, Jan 12 2017
%H A057653 Jean-Christophe Hervé, <a href="/A057653/b057653.txt">Table of n, a(n) for n = 1..4000</a>
%H A057653 Joerg Arndt, <a href="https://arxiv.org/abs/1607.02433">Plane-filling curves on all uniform grids</a>, arXiv preprint arXiv:1607.02433 [math.CO], 2016.
%H A057653 J. H. Conway, E. M. Rains and N. J. A. Sloane, On the existence of similar sublattices, Canad. J. Math. 51 (1999), 1300-1306 (<a href="http://neilsloane.com/doc/sim.txt">Abstract</a>, <a href="http://neilsloane.com/doc/sim.pdf">pdf</a>, <a href="http://neilsloane.com/doc/sim.ps">ps</a>).
%F A057653 n = odd square * {product of distinct primes == 1 (mod 4)}.
%F A057653 a(n) = A097269(n)/2. - _Wolfdieter Lang_, Jan 12 2017
%p A057653 readlib(issqr): for n from 1 to 1001 by 2 do for k from 0 to floor(sqrt(n)) do if issqr(n-k^2) then printf(`%d,`,n); break fi; od:od:
%t A057653 fQ[n_] := Length@ Catch@ Do[If[IntegerQ@ Sqrt[n - k^2], Throw[{k, Sqrt[n - k^2]}], Nothing], {k, Floor[Sqrt@ n]^2}] != 0; Select[Range[1, 340, 2], fQ] (* _Michael De Vlieger_, Nov 13 2015 *)
%o A057653 (PARI) isok(n)=my(f=factor(n)); for(i=1, #f[, 1], if(f[i, 2]%2 && f[i, 1]%4==3, return(0))); 1;
%o A057653 for(n=1, 1e3, if(isok(n) && n%2==1, print1(n", "))) \\ _Altug Alkan_, Nov 13 2015
%o A057653 (PARI) for(n=0, 1e3, if(if( n<1, n==0, 2 * qfrep([ 1, 0; 0, 4], n)[n]) != 0 && n%2==1, print1(n, ", "))) \\ _Altug Alkan_, Nov 19 2015
%o A057653 (Python)
%o A057653 from itertools import count, islice
%o A057653 from sympy import factorint
%o A057653 def A057653_gen(): # generator of terms
%o A057653     return filter(lambda n:all(p & 3 != 3 or e & 1 == 0 for p, e in factorint(n).items()),count(1,2))
%o A057653 A057653_list = list(islice(A057653_gen(),30)) # _Chai Wah Wu_, Jun 28 2022
%Y A057653 Odd members of A001481.
%Y A057653 Odd members of A020668.
%Y A057653 Complement of A084109 in 4k+1 numbers (A016813).
%Y A057653 Cf. A016754 (odd squares), A097269.
%K A057653 nonn
%O A057653 1,2
%A A057653 _N. J. A. Sloane_, Oct 15 2000
%E A057653 More terms from _James Sellers_, Oct 16 2000