%I #6 Jul 31 2021 23:16:53

%S 6112,6138,6462,6497,7001,7038,7057,7064,7099,7190,7316,7328,7372,

%T 7433,7561,7587,7703,7759,7841,7902,8163,8352,8443,8560,8630,8632,

%U 8928,8991,9017,9136,9143,9171,9288,9316,9379,9505,9566,9647,9658,9675,9684,9745,9773

%N Numbers that are the sum of five third powers in exactly nine ways.

%C Differs from A345185 at term 1 because 5860 = 1^3 + 1^3 + 5^3 + 8^3 + 16^3 = 1^3 + 2^3 + 3^3 + 11^3 + 15^3 = 1^3 + 3^3 + 8^3 + 11^3 + 14^3 = 1^3 + 5^3 + 5^3 + 10^3 + 15^3 = 1^3 + 9^3 + 10^3 + 10^3 + 12^3 = 2^3 + 3^3 + 8^3 + 9^3 + 15^3 = 2^3 + 3^3 + 5^3 + 12^3 + 14^3 = 2^3 + 8^3 + 8^3 + 12^3 + 12^3 = 3^3 + 8^3 + 8^3 + 9^3 + 14^3 = 3^3 + 6^3 + 7^3 + 12^3 + 13^3.

%H David Consiglio, Jr., <a href="/A345186/b345186.txt">Table of n, a(n) for n = 1..10000</a>

%e 6112 is a term because 6112 = 1^3 + 2^3 + 9^3 + 11^3 + 14^3 = 1^3 + 3^3 + 7^3 + 12^3 + 14^3 = 1^3 + 6^3 + 6^3 + 7^3 + 16^3 = 2^3 + 2^3 + 9^3 + 9^3 + 15^3 = 2^3 + 3^3 + 5^3 + 11^3 + 15^3 = 2^3 + 8^3 + 9^3 + 9^3 + 14^3 = 3^3 + 3^3 + 3^3 + 4^3 + 17^3 = 3^3 + 5^3 + 8^3 + 11^3 + 14^3 = 8^3 + 8^3 + 8^3 + 11^3 + 12^3.

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

%o for pos in cwr(power_terms, 5):

%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. A294743, A341892, A345154, A345184, A345185, A345188, A345771.

%K nonn

%O 1,1

%A _David Consiglio, Jr._, Jun 10 2021