A345510 - OEIS

%I #12 May 10 2024 08:51:55

%S 34,37,40,42,43,45,46,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,

%T 64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,

%U 87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104

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

%H Sean A. Irvine, <a href="/A345510/b345510.txt">Table of n, a(n) for n = 1..1000</a>

%H a(n) = 2*a(n-1) - a(n-2) for n > 9.

%H G.f.: x*(-x^8 + x^7 - x^6 + x^5 - x^4 - x^3 - 31*x + 34)/(x - 1)^2.

%F From _Chai Wah Wu_, May 09 2024: (Start)

%F 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 a(n) = 2*a(n-1) - a(n-2) for n > 9.

%F G.f.: x*(-x^8 + x^7 - x^6 + x^5 - x^4 - x^3 - 31*x + 34)/(x - 1)^2. (End)

%e 37 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 2^2 + 5^2

%e    = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 2^2 + 3^2 + 3^2 + 3^2

%e    = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 2^2 + 2^2 + 2^2 + 2^2 + 4^2

%e    = 1^2 + 2^2 + 2^2 + 2^2 + 2^2 + 2^2 + 2^2 + 2^2 + 2^2 + 2^2

%e so 37 is a term.

%o (Python)

%o from itertools import combinations_with_replacement as cwr

%o from collections import defaultdict

%o keep = defaultdict(lambda: 0)

%o power_terms = [x**2 for x in range(1, 1000)]

%o for pos in cwr(power_terms, 10):

%o     tot = sum(pos)

%o     keep[tot] += 1

%o     rets = sorted([k for k, v in keep.items() if v >= 3])

%o     for x in range(len(rets)):

%o         print(rets[x])

%Y Cf. A345500, A345509, A345551.

%K nonn

%O 1,1

%A _David Consiglio, Jr._, Jun 20 2021