A155563 Intersection of A001481 and A003136: N = a^2 + b^2 = c^2 + 3d^2 for some integers a,b,c,d.

0, 1, 4, 9, 13, 16, 25, 36, 37, 49, 52, 61, 64, 73, 81, 97, 100, 109, 117, 121, 144, 148, 157, 169, 181, 193, 196, 208, 225, 229, 241, 244, 256, 277, 289, 292, 313, 324, 325, 333, 337, 349, 361, 373, 388, 397, 400, 409, 421, 433, 436, 441, 457, 468, 481, 484

Offset

1,3

Comments

Contains A155561 as a subsequence (obtained by restricting a,b,c,d to be nonzero). Also contains A000290 (squares) as subsequence.

Links

Table of n, a(n) for n=1..56.

Ron Lifshitz, Theory of color symmetry for periodic and quasiperiodic crystals, Rev. Mod. Phys. 69, 1181 (1997). This sequence coincides with the row N = 12 of Table VII.

Prog

(PARI) isA155563(n, /* use optional 2nd arg to get other analogous sequences */c=[3, 1]) = { for(i=1, #c, for(b=0, sqrtint(n\c[i]), issquare(n-c[i]*b^2) & next(2)); return); 1}
for( n=0, 500, isA155563(n) & print1(n", "))
(PARI) is(n)=(n==0) || (#bnfisintnorm(bnfinit(z^2+z+1), n) && #bnfisintnorm(bnfinit(z^2+1), n));
select(n->is(n), vector(1500, j, j-1)) \\ Joerg Arndt, Jan 11 2015
(Python)
from itertools import count, islice
from sympy import factorint
def A155563_gen(): # generator of terms
    return filter(lambda n: all(e & 1 == 0 or (p & 3 != 3 and p % 3 < 2) for p, e in factorint(n).items()), count(0))
A155563_list = list(islice(A155563_gen(), 30)) # Chai Wah Wu, Jun 27 2022

Crossrefs

Cf. A000290, A001481, A003136, A155561.

Sequence in context: A312885 A312886 A010397 * A020684 A312887 A312888

Adjacent sequences: A155560 A155561 A155562 * A155564 A155565 A155566

Keyword

easy,nonn

Author

M. F. Hasler, Jan 24 2009

Status

approved