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

Author

M. F. Hasler, Jan 25 2009

Keywords

• easy
• nonn

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

Crossrefs

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

Programs

Mathematica

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

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