A345497 Numbers that are the sum of eight squares in ten or more ways.

70, 71, 73, 74, 77, 78, 79, 80, 82, 83, 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, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131

Offset

1,1

Links

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

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Formula

From Chai Wah Wu, May 09 2024: (Start)

All integers >= 85 are terms. Proof: since 594 can be written as the sum of 3 positive squares in 10 ways (see A025427) and any integer >= 34 can be written as a sum of 5 positive squares (see A025429), any integer >= 628 can be written as a sum of 8 positive squares in 10 or more ways. Integers from 85 to 627 are terms by inspection.

a(n) = 2*a(n-1) - a(n-2) for n > 12.

G.f.: x*(-x^11 + x^10 - x^9 + x^8 - 2*x^5 + 2*x^4 - x^3 + x^2 - 69*x + 70)/(x - 1)^2. (End)

Example

71 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 8^2
   = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 4^2 + 7^2
   = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 4^2 + 5^2 + 5^2
   = 1^2 + 1^2 + 1^2 + 2^2 + 4^2 + 4^2 + 4^2 + 4^2
   = 1^2 + 1^2 + 1^2 + 3^2 + 3^2 + 3^2 + 4^2 + 5^2
   = 1^2 + 1^2 + 2^2 + 2^2 + 2^2 + 2^2 + 2^2 + 7^2
   = 1^2 + 1^2 + 2^2 + 2^2 + 2^2 + 4^2 + 4^2 + 5^2
   = 1^2 + 2^2 + 2^2 + 2^2 + 2^2 + 2^2 + 5^2 + 5^2
   = 1^2 + 2^2 + 2^2 + 2^2 + 2^2 + 3^2 + 3^2 + 6^2
   = 1^2 + 3^2 + 3^2 + 3^2 + 3^2 + 3^2 + 3^2 + 4^2
   = 2^2 + 2^2 + 2^2 + 3^2 + 3^2 + 3^2 + 4^2 + 4^2
so 71 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, 8):
    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])
(Python)
def A345397(n): return (70, 71, 73, 74, 77, 78, 79, 80, 82, 83)[n-1] if n<11 else n+74 # Chai Wah Wu, May 09 2024

Crossrefs

Cf. A025427, A025429, A345487, A345496, A345540, A346803.

Sequence in context: A245046 A265727 A166506 * A078291 A118216 A256082

Adjacent sequences: A345494 A345495 A345496 * A345498 A345499 A345500

Keyword

nonn

Author

David Consiglio, Jr., Jun 20 2021

Status

approved