%I #6 Jul 31 2021 17:39:18

%S 8259,9299,9539,10709,10819,10884,10949,10964,11124,11444,11573,11668,

%T 11684,11924,12099,12164,12339,12404,12549,12708,12773,12853,12918,

%U 12948,13013,13139,13204,13269,13284,13349,13379,13444,13509,13524,13589,13764,13829

%N Numbers that are the sum of nine fourth powers in nine or more ways.

%H Sean A. Irvine, <a href="/A345593/b345593.txt">Table of n, a(n) for n = 1..10000</a>

%e 9299 is a term because 9299 = 1^4 + 1^4 + 1^4 + 2^4 + 6^4 + 6^4 + 6^4 + 6^4 + 8^4 = 1^4 + 1^4 + 3^4 + 4^4 + 4^4 + 4^4 + 4^4 + 8^4 + 8^4 = 1^4 + 2^4 + 2^4 + 2^4 + 2^4 + 2^4 + 4^4 + 7^4 + 9^4 = 1^4 + 2^4 + 2^4 + 2^4 + 2^4 + 3^4 + 6^4 + 6^4 + 9^4 = 2^4 + 2^4 + 2^4 + 2^4 + 3^4 + 4^4 + 7^4 + 7^4 + 8^4 = 2^4 + 2^4 + 2^4 + 3^4 + 3^4 + 6^4 + 6^4 + 7^4 + 8^4 = 2^4 + 2^4 + 4^4 + 4^4 + 4^4 + 6^4 + 7^4 + 7^4 + 7^4 = 2^4 + 3^4 + 4^4 + 4^4 + 6^4 + 6^4 + 6^4 + 7^4 + 7^4 = 3^4 + 3^4 + 4^4 + 4^4 + 4^4 + 4^4 + 4^4 + 6^4 + 9^4 = 3^4 + 3^4 + 4^4 + 6^4 + 6^4 + 6^4 + 6^4 + 6^4 + 7^4.

%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**4 for x in range(1, 1000)]

%o for pos in cwr(power_terms, 9):

%o tot = sum(pos)

%o keep[tot] += 1

%o rets = sorted([k for k, v in keep.items() if v >= 9])

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

%o print(rets[x])

%Y Cf. A345548, A345584, A345592, A345594, A345602, A345626, A345851.

%K nonn

%O 1,1

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