cp's OEIS Frontend

A345510 Numbers that are the sum of ten squares in three or more ways.

%I A345510 #14 Jun 13 2025 11:25:20
%S A345510 34,37,40,42,43,45,46,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,
%T A345510 64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,
%U A345510 87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104
%N A345510 Numbers that are the sum of ten squares in three or more ways.
%H A345510 Sean A. Irvine, <a href="/A345510/b345510.txt">Table of n, a(n) for n = 1..1000</a>
%H A345510 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F A345510 From _Chai Wah Wu_, May 09 2024: (Start)
%F A345510 All integers >= 48 are terms. Proof: since 29 can be written as the sum of 5 positive squares in 3 ways and any integer >= 34 can be written as a sum of 5 positive squares (see A025429), any integer >= 63 can be written as a sum of 10 positive squares in 3 or more ways. Integers from 48 to 62 are terms by inspection.
%F A345510 a(n) = 2*a(n-1) - a(n-2) for n > 9.
%F A345510 G.f.: x*(-x^8 + x^7 - x^6 + x^5 - x^4 - x^3 - 31*x + 34)/(x - 1)^2. (End)
%e A345510 37 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 2^2 + 5^2
%e A345510    = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 2^2 + 3^2 + 3^2 + 3^2
%e A345510    = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 2^2 + 2^2 + 2^2 + 2^2 + 4^2
%e A345510    = 1^2 + 2^2 + 2^2 + 2^2 + 2^2 + 2^2 + 2^2 + 2^2 + 2^2 + 2^2
%e A345510 so 37 is a term.
%o A345510 (Python)
%o A345510 from itertools import combinations_with_replacement as cwr
%o A345510 from collections import defaultdict
%o A345510 keep = defaultdict(lambda: 0)
%o A345510 power_terms = [x**2 for x in range(1, 1000)]
%o A345510 for pos in cwr(power_terms, 10):
%o A345510     tot = sum(pos)
%o A345510     keep[tot] += 1
%o A345510     rets = sorted([k for k, v in keep.items() if v >= 3])
%o A345510     for x in range(len(rets)):
%o A345510         print(rets[x])
%Y A345510 Cf. A345500, A345509, A345551.
%K A345510 nonn
%O A345510 1,1
%A A345510 _David Consiglio, Jr._, Jun 20 2021