A345476 - OEIS

%I #15 May 10 2024 01:39:59

%S 78,81,84,86,89,92,93,95,99,100,101,102,104,105,107,108,110,111,113,

%T 114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,

%U 131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148

%N Numbers that are the sum of six squares in nine or more ways.

%H Sean A. Irvine, <a href="/A345476/b345476.txt">Table of n, a(n) for n = 1..1000</a>

%F Conjectures from _Chai Wah Wu_, Jan 05 2024: (Start)

%F a(n) = 2*a(n-1) - a(n-2) for n > 20.

%F G.f.: x*(-x^19 + x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - 3*x^9 + 2*x^8 + x^7 - 2*x^6 + x^4 - x^3 - 75*x + 78)/(x - 1)^2. (End)

%e 81 = 1^2 + 1^2 + 1^2 + 2^2 + 5^2 + 7^2

%e    = 1^2 + 1^2 + 2^2 + 5^2 + 5^2 + 5^2

%e    = 1^2 + 1^2 + 3^2 + 3^2 + 5^2 + 6^2

%e    = 1^2 + 2^2 + 2^2 + 2^2 + 2^2 + 8^2

%e    = 1^2 + 2^2 + 3^2 + 3^2 + 3^2 + 7^2

%e    = 1^2 + 4^2 + 4^2 + 4^2 + 4^2 + 4^2

%e    = 2^2 + 2^2 + 2^2 + 2^2 + 4^2 + 7^2

%e    = 2^2 + 2^2 + 4^2 + 4^2 + 4^2 + 5^2

%e    = 2^2 + 3^2 + 3^2 + 3^2 + 5^2 + 5^2

%e    = 3^2 + 3^2 + 3^2 + 3^2 + 3^2 + 6^2

%e so 81 is a term.

%o (Python)

%o from itertools import combinations_with_replacement as cwr

%o from collections import defaultdict

%o keep = defaultdict(lambda: 0)

%o power_terms = [x**2 for x in range(1, 1000)]

%o for pos in cwr(power_terms, 6):

%o     tot = sum(pos)

%o     keep[tot] += 1

%o     rets = sorted([k for k, v in keep.items() if v >= 9])

%o     for x in range(len(rets)):

%o         print(rets[x])

%Y Cf. A344802, A344812, A345477, A345486, A345518.

%K nonn

%O 1,1

%A _David Consiglio, Jr._, Jun 20 2021