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%I #6 Jul 31 2021 21:33:48
%S 13268,14212,14788,15667,16612,16627,16707,16772,16822,16852,16882,
%T 16947,17363,17428,17877,18117,18948,19157,19237,19252,19682,19828,
%U 20291,20372,20612,20707,20722,20772,20917,20962,21253,21331,21458,21478,21573,21717,21763
%N Numbers that are the sum of eight fourth powers in exactly eight ways.
%C Differs from A345583 at term 4 because 15427 = 1^4 + 1^4 + 1^4 + 2^4 + 2^4 + 6^4 + 8^4 + 10^4 = 1^4 + 2^4 + 2^4 + 2^4 + 5^4 + 8^4 + 8^4 + 9^4 = 1^4 + 4^4 + 5^4 + 6^4 + 6^4 + 6^4 + 8^4 + 9^4 = 2^4 + 2^4 + 2^4 + 2^4 + 2^4 + 3^4 + 5^4 + 11^4 = 2^4 + 2^4 + 3^4 + 4^4 + 4^4 + 7^4 + 7^4 + 10^4 = 2^4 + 2^4 + 3^4 + 5^4 + 7^4 + 8^4 + 8^4 + 8^4 = 2^4 + 3^4 + 3^4 + 4^4 + 6^4 + 6^4 + 7^4 + 10^4 = 3^4 + 3^4 + 3^4 + 6^4 + 6^4 + 6^4 + 6^4 + 10^4 = 4^4 + 4^4 + 5^4 + 6^4 + 7^4 + 7^4 + 8^4 + 8^4.
%H Sean A. Irvine, <a href="/A345840/b345840.txt">Table of n, a(n) for n = 1..10000</a>
%e 14212 is a term because 14212 = 1^4 + 1^4 + 1^4 + 2^4 + 2^4 + 3^4 + 8^4 + 10^4 = 1^4 + 1^4 + 1^4 + 4^4 + 4^4 + 6^4 + 7^4 + 10^4 = 1^4 + 1^4 + 1^4 + 5^4 + 6^4 + 8^4 + 8^4 + 8^4 = 1^4 + 2^4 + 4^4 + 4^4 + 5^4 + 7^4 + 8^4 + 9^4 = 1^4 + 3^4 + 4^4 + 5^4 + 6^4 + 6^4 + 8^4 + 9^4 = 2^4 + 3^4 + 3^4 + 3^4 + 4^4 + 6^4 + 7^4 + 10^4 = 3^4 + 3^4 + 3^4 + 3^4 + 6^4 + 6^4 + 6^4 + 10^4 = 3^4 + 4^4 + 4^4 + 5^4 + 7^4 + 7^4 + 8^4 + 8^4.
%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, 8):
%o tot = sum(pos)
%o keep[tot] += 1
%o rets = sorted([k for k, v in keep.items() if v == 8])
%o for x in range(len(rets)):
%o print(rets[x])
%Y Cf. A345583, A345790, A345830, A345839, A345841, A345850, A346333.
%K nonn
%O 1,1
%A _David Consiglio, Jr._, Jun 26 2021
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