A155717 Numbers of the form N = a^2 + 7b^2 for some positive integers a,b.

8, 11, 16, 23, 29, 32, 37, 43, 44, 53, 56, 64, 67, 71, 72, 77, 79, 88, 92, 99, 107, 109, 112, 113, 116, 121, 127, 128, 137, 144, 148, 149, 151, 161, 163, 172, 176, 179, 184, 191, 193, 197, 200, 203, 207, 211, 212, 224, 232, 233, 239, 253, 256, 259, 261, 263, 268

Offset

1,1

Comments

Subsequence of A020670 (which allows for a and b to be zero).

If N=a^2+7*b^2 is a term then 7*N=(7*b)^2+7*a^2 is also a term. Conversely,if 7*N is a term then N is a term. Example: N=56; N/7=8 is a term, N*7=7^2+7*7^2 is a term. Sequences A154777, A092572 and A154778 have the same property with 7 replaced by prime numbers 2,3 and 5 respectively. - Jerzy R Borysowicz, May 22 2020

Links

Table of n, a(n) for n=1..57.

Mathematica

Select[Range[300], Reduce[a>0 && b>0 && # == a^2 + 7b^2, {a, b}, Integers] =!= False&] (* Jean-François Alcover, Nov 17 2016 *)

Prog

(PARI) isA155717(n, /* optional 2nd arg allows us to get other sequences */c=7) = { for(b=1, sqrtint((n-1)\c), issquare(n-c*b^2) & return(1))}
for( n=1, 300, isA155717(n) & print1(n", "))
(Python)
def aupto(limit):
    cands = range(1, int(limit**.5)+2)
    nums = [a**2 + 7*b**2 for a in cands for b in cands]
    return sorted(set(k for k in nums if k <= limit))
print(aupto(268)) # Michael S. Branicky, Aug 11 2021

Crossrefs

Cf. A000404, A154777, A092572, A097268, A154778, A155707-A155716, A155560-A155578.

Sequence in context: A154685 A068591 A065348 * A188197 A234096 A111254

Adjacent sequences: A155714 A155715 A155716 * A155718 A155719 A155720

Keyword

easy,nonn

Author

M. F. Hasler, Jan 25 2009

Status

approved