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

1,1

Links

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

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)

Example

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.

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

Crossrefs

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

Sequence in context: A183984 A089784 A186464 * A295700 A286898 A184044

Adjacent sequences: A345474 A345475 A345476 * A345478 A345479 A345480

Keyword

nonn

Author

David Consiglio, Jr., Jun 20 2021

Status

approved