A345477 Numbers that are the sum of six squares in ten or more ways.

81, 84, 86, 89, 92, 93, 95, 100, 101, 102, 104, 105, 107, 108, 110, 111, 113, 114, 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, 143, 144, 145, 146, 147, 148, 149, 150

Offset: 1

Author

David Consiglio, Jr., Jun 20 2021

Keywords

• nonn

Examples

			84 = 1^2 + 1^2 + 1^2 + 1^2 + 4^2 + 8^2
   = 1^2 + 1^2 + 1^2 + 3^2 + 6^2 + 6^2
   = 1^2 + 1^2 + 1^2 + 4^2 + 4^2 + 7^2
   = 1^2 + 1^2 + 2^2 + 2^2 + 5^2 + 7^2
   = 1^2 + 1^2 + 4^2 + 4^2 + 5^2 + 5^2
   = 1^2 + 2^2 + 2^2 + 5^2 + 5^2 + 5^2
   = 1^2 + 2^2 + 3^2 + 3^2 + 5^2 + 6^2
   = 2^2 + 2^2 + 2^2 + 2^2 + 2^2 + 8^2
   = 2^2 + 2^2 + 3^2 + 3^2 + 3^2 + 7^2
   = 2^2 + 4^2 + 4^2 + 4^2 + 4^2 + 4^2
   = 3^2 + 3^2 + 3^2 + 4^2 + 4^2 + 5^2
so 84 is a term.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from Sean A. Irvine)

Crossrefs

Cf. A025430, A344803, A345476, A345487, A345519.

Programs

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 >= 10])
    for x in range(len(rets)):
        print(rets[x])

Formula

Conjectures from Chai Wah Wu, Jan 05 2024: (Start)

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

G.f.: x*(-x^19 + x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - 4*x^8 + 3*x^7 + x^6 - 2*x^5 + x^3 - x^2 - 78*x + 81)/(x - 1)^2. (End)