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A345809 Numbers that are the sum of ten cubes in exactly seven ways.

%I A345809 #6 Jul 31 2021 22:27:30
%S A345809 440,473,499,506,525,532,534,567,571,584,588,597,599,604,606,627,637,
%T A345809 639,640,656,660,663,669,670,673,680,682,689,691,693,696,701,702,704,
%U A345809 707,717,718,719,726,729,735,738,743,744,750,755,761,762,763,770,783,784
%N A345809 Numbers that are the sum of ten cubes in exactly seven ways.
%C A345809 Differs from A345555 at term 16 because 623 = 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 3^3 + 5^3 + 5^3 + 5^3 + 6^3  = 1^3 + 1^3 + 1^3 + 1^3 + 2^3 + 3^3 + 3^3 + 5^3 + 6^3 + 6^3  = 1^3 + 1^3 + 1^3 + 2^3 + 2^3 + 2^3 + 4^3 + 4^3 + 5^3 + 7^3  = 1^3 + 1^3 + 3^3 + 3^3 + 4^3 + 4^3 + 4^3 + 5^3 + 5^3 + 5^3  = 1^3 + 2^3 + 3^3 + 3^3 + 3^3 + 4^3 + 4^3 + 4^3 + 5^3 + 6^3  = 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 6^3 + 7^3  = 2^3 + 2^3 + 2^3 + 2^3 + 3^3 + 4^3 + 5^3 + 5^3 + 5^3 + 5^3  = 3^3 + 3^3 + 3^3 + 3^3 + 3^3 + 3^3 + 3^3 + 3^3 + 4^3 + 7^3.
%C A345809 Likely finite.
%H A345809 Sean A. Irvine, <a href="/A345809/b345809.txt">Table of n, a(n) for n = 1..78</a>
%e A345809 473 is a term because 473 = 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 4^3 + 4^3 + 5^3 = 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 2^3 + 3^3 + 5^3 + 5^3 = 1^3 + 1^3 + 1^3 + 1^3 + 3^3 + 3^3 + 3^3 + 3^3 + 4^3 + 4^3 = 1^3 + 1^3 + 1^3 + 2^3 + 3^3 + 3^3 + 3^3 + 3^3 + 3^3 + 5^3 = 1^3 + 1^3 + 2^3 + 2^3 + 2^3 + 2^3 + 3^3 + 4^3 + 4^3 + 4^3 = 1^3 + 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 3^3 + 3^3 + 4^3 + 5^3 = 2^3 + 3^3 + 3^3 + 3^3 + 3^3 + 3^3 + 3^3 + 3^3 + 3^3 + 3^3.
%o A345809 (Python)
%o A345809 from itertools import combinations_with_replacement as cwr
%o A345809 from collections import defaultdict
%o A345809 keep = defaultdict(lambda: 0)
%o A345809 power_terms = [x**3 for x in range(1, 1000)]
%o A345809 for pos in cwr(power_terms, 10):
%o A345809     tot = sum(pos)
%o A345809     keep[tot] += 1
%o A345809     rets = sorted([k for k, v in keep.items() if v == 7])
%o A345809     for x in range(len(rets)):
%o A345809         print(rets[x])
%Y A345809 Cf. A345555, A345799, A345808, A345810, A345859.
%K A345809 nonn
%O A345809 1,1
%A A345809 _David Consiglio, Jr._, Jun 26 2021