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

1, 3, 7, 13, 19, 31, 37, 43, 49, 61, 67, 73, 79, 91, 111, 127, 163, 169, 183, 199, 223, 307, 313, 349, 361, 397, 433, 511, 523, 541, 613, 619, 709, 823, 907, 1087, 1123, 1129, 1147, 1213, 1279, 1434

Offset: 1

Author

V. Raman, Aug 26 2012

Keywords

• nonn
• more

Comments

These Lucas numbers L(n) have no prime factor congruent to 3 mod 4 to an odd power.

Also, numbers n such that L(n) can be written in the form a^2 + 5*b^2.

Subsequence of A124132.

Is this A124132 without the 6? - Joerg Arndt, Sep 07 2012

Any prime factor of Lucas(n) for the prime values of n is always of the form 1 (mod 10) or 9 (mod 10).

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.

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

Examples

			Lucas(19) = 9349 = 95^2 + 18^2.
Lucas(19) = 9349 = 23^2 + 5*42^2.
		

Links

• Blair Kelly, Fibonacci and Lucas factorizations
• Eric W. Weisstein, Sum of squares function

Crossrefs

Cf. A000032, A124130, A215809, A215906.

Cf. A180363.

Cf. A020669, A033205 (numbers and primes of the form x^2 + 5*y^2).

Programs

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.

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.

Extensions

17 more terms from V. Raman, Aug 28 2012

A215940 merged into this sequence by T. D. Noe, Sep 21 2012

a(38)-a(41) from Chai Wah Wu, Jul 22 2020