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A345482
Numbers that are the sum of seven squares in five or more ways.
6
45, 54, 55, 57, 58, 60, 61, 63, 64, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117
OFFSET
1,1
LINKS
FORMULA
Conjectures from Chai Wah Wu, Apr 25 2024: (Start)
a(n) = 2*a(n-1) - a(n-2) for n > 13.
G.f.: x*(-x^12 + x^11 - x^10 + x^9 - x^8 + x^7 - x^6 + x^5 - x^4 + x^3 - 8*x^2 - 36*x + 45)/(x - 1)^2. (End)
EXAMPLE
54 = 1^2 + 1^2 + 1^2 + 1^2 + 3^2 + 4^2 + 5^2
= 1^2 + 1^2 + 2^2 + 2^2 + 2^2 + 2^2 + 6^2
= 1^2 + 1^2 + 3^2 + 3^2 + 3^2 + 3^2 + 4^2
= 1^2 + 2^2 + 2^2 + 2^2 + 3^2 + 4^2 + 4^2
= 2^2 + 2^2 + 2^2 + 2^2 + 2^2 + 3^2 + 5^2
so 54 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, 7):
tot = sum(pos)
keep[tot] += 1
rets = sorted([k for k, v in keep.items() if v >= 5])
for x in range(len(rets)):
print(rets[x])
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved