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A155717
Numbers of the form N = a^2 + 7b^2 for some positive integers a,b.
11
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
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
KEYWORD
easy,nonn
AUTHOR
M. F. Hasler, Jan 25 2009
STATUS
approved