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Numbers that are the sum of five positive fifth powers in exactly four ways.
6

%I #10 Jan 11 2025 04:14:29

%S 287618651,1386406515,1763135232,2494769760,2619898293,3096064443,

%T 3291315732,3749564512,4045994624,5142310350,5183605813,5658934676,

%U 5880926107,7205217018,7401155424,7691215599,8429499101,8926086432,9051501568,9203796832,9254212901

%N Numbers that are the sum of five positive fifth powers in exactly four ways.

%C Differs from A344518 at term 20 because

%C 9006349824 = 8^5 + 34^5 + 62^5 + 68^5 + 92^5

%C = 8^5 + 41^5 + 47^5 + 79^5 + 89^5

%C = 12^5 + 18^5 + 72^5 + 78^5 + 84^5

%C = 21^5 + 34^5 + 43^5 + 74^5 + 92^5

%C = 24^5 + 42^5 + 48^5 + 54^5 + 96^5.

%H Sean A. Irvine, <a href="/A344519/b344519.txt">Table of n, a(n) for n = 1..4857</a>

%e 287618651 is a term because

%e 287618651 = 8^5 + 21^5 + 27^5 + 27^5 + 48^5

%e = 9^5 + 13^5 + 26^5 + 37^5 + 46^5

%e = 11^5 + 12^5 + 23^5 + 41^5 + 44^5

%e = 11^5 + 20^5 + 22^5 + 30^5 + 48^5.

%e [Corrected by _Patrick De Geest_, Dec 28 2024]

%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**5 for x in range(1, 500)]

%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 == 4])

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

%o print(rets[x])

%Y Cf. A003350, A004845, A342685, A342686, A342687, A342688, A344244, A344355, A344518, A345863, A345864, A346257, A346358, A346359.

%K nonn

%O 1,1

%A _David Consiglio, Jr._, May 21 2021