A155578 Intersection of A000404 and A155717: N = a^2 + b^2 = c^2 + 7*d^2 for some positive integers a,b,c,d.
8, 29, 32, 37, 53, 72, 109, 113, 116, 128, 137, 148, 149, 193, 197, 200, 212, 232, 233, 261, 277, 281, 288, 296, 317, 333, 337, 373, 389, 392, 400, 401, 421, 424, 436, 449, 452, 457, 464, 477, 512, 541, 548, 557, 569, 592, 596, 613, 617, 641, 648, 653, 673
Offset: 1
Programs
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PARI
isA155578(n,/* optional 2nd arg allows us to get other sequences */c=[7,1]) = { for(i=1,#c, for(b=1,sqrtint((n-1)\c[i]), issquare(n-c[i]*b^2) & next(2)); return);1} for( n=1,999, isA155578(n) & print1(n","))
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Python
from math import isqrt def aupto(limit): cands = range(1, isqrt(limit)+1) left = set(a**2 + b**2 for a in cands for b in cands) right = set(c**2 + 7*d**2 for c in cands for d in cands) return sorted(k for k in left & right if k <= limit) print(aupto(673)) # Michael S. Branicky, Aug 29 2021
Comments