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A155717
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Numbers of the form N = a^2 + 7b^2 for some positive integers a,b.
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11
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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MATHEMATICA
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Select[Range[300], Reduce[a>0 && b>0 && # == a^2 + 7b^2, {a, b}, Integers] =!= False&] (* Jean-François Alcover, Nov 17 2016 *)
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PROG
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(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))
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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