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A345822 Numbers that are the sum of six fourth powers in exactly ten ways. 7

%I #6 Jul 31 2021 21:57:40

%S 122915,151556,161475,162755,173075,183620,185315,199106,199940,

%T 201875,202275,204275,204340,204595,206115,207395,209795,211075,

%U 213731,217826,217891,218515,221250,223955,224180,225875,226595,227186,228035,236195,237796,237890

%N Numbers that are the sum of six fourth powers in exactly ten ways.

%C Differs from A345567 at term 8 because 197795 = 1^4 + 2^4 + 5^4 + 6^4 + 16^4 + 19^4 = 1^4 + 2^4 + 7^4 + 11^4 + 12^4 + 20^4 = 1^4 + 2^4 + 10^4 + 12^4 + 17^4 + 17^4 = 2^4 + 4^4 + 7^4 + 9^4 + 13^4 + 20^4 = 2^4 + 11^4 + 13^4 + 14^4 + 15^4 + 16^4 = 3^4 + 6^4 + 6^4 + 9^4 + 13^4 + 20^4 = 3^4 + 6^4 + 7^4 + 14^4 + 15^4 + 18^4 = 4^4 + 9^4 + 11^4 + 12^4 + 15^4 + 18^4 = 7^4 + 7^4 + 14^4 + 14^4 + 15^4 + 16^4.

%H Sean A. Irvine, <a href="/A345822/b345822.txt">Table of n, a(n) for n = 1..10000</a>

%e 151556 is a term because 151556 = 1^4 + 2^4 + 2^4 + 9^4 + 11^4 + 19^4 = 1^4 + 2^4 + 3^4 + 7^4 + 16^4 + 17^4 = 1^4 + 8^4 + 11^4 + 12^4 + 13^4 + 17^4 = 2^4 + 3^4 + 7^4 + 8^4 + 11^4 + 19^4 = 3^4 + 3^4 + 3^4 + 4^4 + 12^4 + 19^4 = 3^4 + 4^4 + 11^4 + 11^4 + 14^4 + 17^4 = 3^4 + 4^4 + 13^4 + 13^4 + 13^4 + 16^4 = 4^4 + 6^4 + 9^4 + 9^4 + 9^4 + 19^4 = 4^4 + 7^4 + 11^4 + 11^4 + 11^4 + 18^4 = 4^4 + 8^4 + 9^4 + 13^4 + 13^4 + 17^4.

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

%o for pos in cwr(power_terms, 6):

%o tot = sum(pos)

%o keep[tot] += 1

%o rets = sorted([k for k, v in keep.items() if v == 10])

%o for x in range(len(rets)):

%o print(rets[x])

%Y Cf. A341898, A345567, A345772, A345821, A345832, A346365.

%K nonn

%O 1,1

%A _David Consiglio, Jr._, Jun 26 2021

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