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A343704 Numbers that are the sum of five positive cubes in three or more ways.

%I A343704 #24 Jul 29 2023 13:55:12
%S A343704 766,810,827,829,865,883,981,1018,1025,1044,1070,1105,1108,1142,1145,
%T A343704 1161,1168,1226,1233,1252,1259,1289,1350,1368,1376,1424,1431,1439,
%U A343704 1441,1457,1461,1487,1492,1494,1522,1529,1531,1538,1548,1550,1555,1568,1583,1585,1587,1590,1592,1593,1594,1609,1611,1613,1639
%N A343704 Numbers that are the sum of five positive cubes in three or more ways.
%C A343704 This sequence differs from A343705 at term 20 because 1252 = 1^3+1^3+5^3+5^3+10^3= 1^3+2^3+3^3+6^3+10^3 = 3^3+3^3+7^3+7^3+8^3 = 3^3+4^3+6^3+6^3+9^3. Thus this term is in this sequence but not A343705.
%H A343704 David Consiglio, Jr., <a href="/A343704/b343704.txt">Table of n, a(n) for n = 1..10000</a>
%e A343704 827 is a member of this sequence because 827 = 1^3 + 4^3 + 5^3 + 5^3 + 8^3 = 2^3 + 2^3 + 5^3 + 7^3 + 7^3 = 2^3 + 3^3 + 4^3 + 6^3 + 8^3.
%t A343704 Select[Range@2000,Length@Select[PowersRepresentations[#,5,3],FreeQ[#,0]&]>2&] (* _Giorgos Kalogeropoulos_, Apr 26 2021 *)
%o A343704 (Python)
%o A343704 from itertools import combinations_with_replacement as cwr
%o A343704 from collections import defaultdict
%o A343704 keep = defaultdict(lambda: 0)
%o A343704 power_terms = [x**3 for x in range(1,50)]#n
%o A343704 for pos in cwr(power_terms,5):#m
%o A343704     tot = sum(pos)
%o A343704     keep[tot] += 1
%o A343704 rets = sorted([k for k,v in keep.items() if v >= 3])#s
%o A343704 for x in range(len(rets)):
%o A343704     print(rets[x])
%Y A343704 Cf. A025407, A343702, A343705, A344034, A344243, A344796, A345512.
%K A343704 nonn,easy
%O A343704 1,1
%A A343704 _David Consiglio, Jr._, Apr 26 2021