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A344812
Numbers that are the sum of six squares in eight or more ways.
6
78, 81, 84, 86, 87, 89, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 107, 108, 109, 110, 111, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142
OFFSET
1,1
LINKS
FORMULA
Conjectures from Chai Wah Wu, Jan 05 2024: (Start)
a(n) = 2*a(n-1) - a(n-2) for n > 27.
G.f.: x*(-x^26 + x^25 - x^21 + x^20 - 2*x^7 + x^6 + x^5 - x^4 - x^3 - 75*x + 78)/(x - 1)^2. (End)
EXAMPLE
81 = 1^2 + 1^2 + 1^2 + 2^2 + 5^2 + 7^2
= 1^2 + 1^2 + 2^2 + 5^2 + 5^2 + 5^2
= 1^2 + 1^2 + 3^2 + 3^2 + 5^2 + 6^2
= 1^2 + 2^2 + 2^2 + 2^2 + 2^2 + 8^2
= 1^2 + 2^2 + 3^2 + 3^2 + 3^2 + 7^2
= 1^2 + 4^2 + 4^2 + 4^2 + 4^2 + 4^2
= 2^2 + 2^2 + 2^2 + 2^2 + 4^2 + 7^2
= 2^2 + 2^2 + 4^2 + 4^2 + 4^2 + 5^2
= 2^2 + 3^2 + 3^2 + 3^2 + 5^2 + 5^2
= 3^2 + 3^2 + 3^2 + 3^2 + 3^2 + 6^2
so 81 is a term.
PROG
(Python)
from itertools import combinations_with_replacement as cwr
from collections import defaultdict
keep = defaultdict(lambda: 0)
power_terms = [x**2 for x in range(1, 1000)]
for pos in cwr(power_terms, 6):
tot = sum(pos)
keep[tot] += 1
rets = sorted([k for k, v in keep.items() if v >= 8])
for x in range(len(rets)):
print(rets[x])
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved