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

1,2

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

Links

Table of n, a(n) for n=1..42.

Blair Kelly, Fibonacci and Lucas factorizations

Eric W. Weisstein, Sum of squares function

Example

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

Prog

(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.

Crossrefs

Cf. A000032, A124130, A215809, A215906.

Cf. A180363.

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

Sequence in context: A086148 A262086 A205956 * A007645 A144919 A215801

Adjacent sequences: A215904 A215905 A215906 * A215908 A215909 A215910

Keyword

nonn,more

Author

V. Raman, Aug 26 2012

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

Status

approved