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A345816 Numbers that are the sum of six fourth powers in exactly four ways.

%I A345816 #8 Mar 11 2023 14:04:28
%S A345816 6626,6691,6866,9251,9491,10115,10706,10786,11555,12595,14225,14691,
%T A345816 14771,15315,15330,15570,16051,16595,16660,16675,16850,17090,17091,
%U A345816 17236,17316,17331,17346,17860,17875,17940,17955,18195,18786,18851,19155,19170,19475,19490
%N A345816 Numbers that are the sum of six fourth powers in exactly four ways.
%C A345816 Differs from A345561 at term 16 because 15395 = 1^4 + 1^4 + 1^4 + 6^4 + 8^4 + 10^4  = 1^4 + 2^4 + 5^4 + 8^4 + 8^4 + 9^4  = 3^4 + 4^4 + 4^4 + 7^4 + 7^4 + 10^4  = 3^4 + 5^4 + 7^4 + 8^4 + 8^4 + 8^4 = 2^4 + 2^4 + 2^4 + 3^4 + 5^4 + 11^4.
%H A345816 Sean A. Irvine, <a href="/A345816/b345816.txt">Table of n, a(n) for n = 1..10000</a>
%e A345816 6691 is a term because 6691 = 1^4 + 1^4 + 1^4 + 6^4 + 6^4 + 8^4 = 1^4 + 2^4 + 2^4 + 2^4 + 3^4 + 9^4 = 2^4 + 2^4 + 3^4 + 3^4 + 7^4 + 8^4 = 3^4 + 4^4 + 4^4 + 6^4 + 7^4 + 7^4.
%t A345816 Select[Range[20000],Count[PowersRepresentations[#,6,4],_?(#[[1]]>0&)]==4&] (* _Harvey P. Dale_, Mar 11 2023 *)
%o A345816 (Python)
%o A345816 from itertools import combinations_with_replacement as cwr
%o A345816 from collections import defaultdict
%o A345816 keep = defaultdict(lambda: 0)
%o A345816 power_terms = [x**4 for x in range(1, 1000)]
%o A345816 for pos in cwr(power_terms, 6):
%o A345816     tot = sum(pos)
%o A345816     keep[tot] += 1
%o A345816     rets = sorted([k for k, v in keep.items() if v == 4])
%o A345816     for x in range(len(rets)):
%o A345816         print(rets[x])
%Y A345816 Cf. A344355, A345561, A345766, A345815, A345817, A345826, A346359.
%K A345816 nonn
%O A345816 1,1
%A A345816 _David Consiglio, Jr._, Jun 26 2021