cp's OEIS Frontend

A215907 Odd numbers n such that the Lucas number L(n) is the sum of two squares.

%I A215907 #38 Feb 16 2025 08:33:18
%S A215907 1,3,7,13,19,31,37,43,49,61,67,73,79,91,111,127,163,169,183,199,223,
%T A215907 307,313,349,361,397,433,511,523,541,613,619,709,823,907,1087,1123,
%U A215907 1129,1147,1213,1279,1434
%N A215907 Odd numbers n such that the Lucas number L(n) is the sum of two squares.
%C A215907 These Lucas numbers L(n) have no prime factor congruent to 3 mod 4 to an odd power.
%C A215907 Also, numbers n such that L(n) can be written in the form a^2 + 5*b^2.
%C A215907 Subsequence of A124132.
%C A215907 Is this A124132 without the 6? - _Joerg Arndt_, Sep 07 2012
%C A215907 Any prime factor of Lucas(n) for the prime values of n is always of the form 1 (mod 10) or 9 (mod 10).
%C A215907 A number n can be written in the form a^2 + 5*b^2 if and only if n is 0, or of the form 2^(2i) 5^j Product_{p==1 or 9 mod 20} p^k Product_{q==3 or 7 mod 20) q^(2m) or of the form 2^(2i+1) 5^j Product_{p==1 or 9 mod 20} p^k Product_{q==3 or 7 mod 20) q^(2m+1), for integers i,j,k,m, for primes p,q.
%C A215907 1501 <= a(42) <= 1531. 1531, 1651, 1747, 1849, 1951, 2053, 2413, 2449, 2467, 4069, 5107, 5419, 5851, 7243, 7741, 8467, 13963, 14449, 14887, 15511, 15907, 35449, 51169, 193201, 344293, 387433, 574219, 901657, 1051849 are terms. - _Chai Wah Wu_, Jul 22 2020
%H A215907 Blair Kelly, <a href="http://mersennus.net/fibonacci">Fibonacci and Lucas factorizations</a>
%H A215907 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/SumofSquaresFunction.html">Sum of squares function</a>
%e A215907 Lucas(19) = 9349 = 95^2 + 18^2.
%e A215907 Lucas(19) = 9349 = 23^2 + 5*42^2.
%o A215907 (PARI) for(i=2, 500, a=factorint(fibonacci(i-1)+fibonacci(i+1))~; has=0; for(j=1, #a, if(a[1, j]%4==3&&a[2, j]%2==1, has=1; break)); if(has==0&&i%2==1, print(i", "))) \\ a^2 + b^2 form.
%o A215907 (PARI) for(i=2, 500, a=factorint(fibonacci(i-1)+fibonacci(i+1))~; flag=0; flip=0; for(j=1, #a, if(((a[1, j]%20>10))&&a[2, j]%2==1, flag=1); if(((a[1, j]%20==2)||(a[1, j]%20==3)||(a[1, j]%20==7))&&a[2, j]%2==1, flip=flip+1)); if(flag==0&&flip%2==0, print(i", "))) \\ a^2 + 5*b^2 form.
%Y A215907 Cf. A000032, A124130, A215809, A215906.
%Y A215907 Cf. A180363.
%Y A215907 Cf. A020669, A033205 (numbers and primes of the form x^2 + 5*y^2).
%K A215907 nonn,more
%O A215907 1,2
%A A215907 _V. Raman_, Aug 26 2012
%E A215907 17 more terms from _V. Raman_, Aug 28 2012
%E A215907 A215940 merged into this sequence by _T. D. Noe_, Sep 21 2012
%E A215907 a(38)-a(41) from _Chai Wah Wu_, Jul 22 2020