A345487 Numbers that are the sum of seven squares in ten or more ways.

70, 79, 82, 85, 87, 88, 90, 91, 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, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139

Offset

1,1

Links

Sean A. Irvine, Table of n, a(n) for n = 1..1000

Formula

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

a(n) = 2*a(n-1) - a(n-2) for n > 10.

G.f.: x*(-x^9 + x^8 - x^7 + x^6 - x^5 - x^4 - 6*x^2 - 61*x + 70)/(x - 1)^2. (End)

Example

79 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 5^2 + 7^2
   = 1^2 + 1^2 + 1^2 + 1^2 + 5^2 + 5^2 + 5^2
   = 1^2 + 1^2 + 1^2 + 2^2 + 2^2 + 2^2 + 8^2
   = 1^2 + 1^2 + 1^2 + 3^2 + 3^2 + 3^2 + 7^2
   = 1^2 + 1^2 + 2^2 + 2^2 + 2^2 + 4^2 + 7^2
   = 1^2 + 1^2 + 2^2 + 4^2 + 4^2 + 4^2 + 5^2
   = 1^2 + 1^2 + 3^2 + 3^2 + 3^2 + 5^2 + 5^2
   = 1^2 + 2^2 + 2^2 + 2^2 + 4^2 + 5^2 + 5^2
   = 1^2 + 2^2 + 2^2 + 3^2 + 3^2 + 4^2 + 6^2
   = 2^2 + 3^2 + 3^2 + 3^2 + 4^2 + 4^2 + 4^2
   = 3^2 + 3^2 + 3^2 + 3^2 + 3^2 + 3^2 + 5^2
so 79 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 >= 10])
    for x in range(len(rets)):
        print(rets[x])

Crossrefs

Cf. A345477, A345486, A345497, A345506.

Sequence in context: A078291 A118216 A256082 * A114838 A036191 A165632

Adjacent sequences: A345484 A345485 A345486 * A345488 A345489 A345490

Keyword

nonn

Author

David Consiglio, Jr., Jun 20 2021

Status

approved