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A345854 Numbers that are the sum of ten fourth powers in exactly two ways.

%I A345854 #6 Jul 31 2021 20:00:07
%S A345854 265,280,295,310,325,340,345,355,360,375,390,405,420,425,440,455,470,
%T A345854 485,505,565,580,585,595,630,645,660,665,695,710,725,745,760,805,820,
%U A345854 835,840,870,885,889,900,904,919,920,934,935,949,950,964,965,969,984,999
%N A345854 Numbers that are the sum of ten fourth powers in exactly two ways.
%C A345854 Differs from A345595 at term 20 because 520 = 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 4^4 + 4^4  = 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 2^4 + 3^4 + 3^4 + 3^4 + 4^4  = 1^4 + 1^4 + 2^4 + 2^4 + 3^4 + 3^4 + 3^4 + 3^4 + 3^4 + 3^4.
%H A345854 Sean A. Irvine, <a href="/A345854/b345854.txt">Table of n, a(n) for n = 1..10000</a>
%e A345854 280 is a term because 280 = 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 2^4 + 4^4 = 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 2^4 + 2^4 + 3^4 + 3^4 + 3^4.
%o A345854 (Python)
%o A345854 from itertools import combinations_with_replacement as cwr
%o A345854 from collections import defaultdict
%o A345854 keep = defaultdict(lambda: 0)
%o A345854 power_terms = [x**4 for x in range(1, 1000)]
%o A345854 for pos in cwr(power_terms, 10):
%o A345854     tot = sum(pos)
%o A345854     keep[tot] += 1
%o A345854     rets = sorted([k for k, v in keep.items() if v == 2])
%o A345854     for x in range(len(rets)):
%o A345854         print(rets[x])
%Y A345854 Cf. A345595, A345804, A345844, A345853, A345855, A346347.
%K A345854 nonn
%O A345854 1,1
%A A345854 _David Consiglio, Jr._, Jun 26 2021