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Numbers that are the sum of four fourth powers in exactly nine ways.
7

%I #6 Jul 31 2021 22:12:24

%S 328118259,385202034,395613234,489597858,625839858,641398338,

%T 674511618,693239634,699598578,722302434,779889314,780278643,

%U 782999714,791204514,792005379,797405714,797935698,805299699,815120658,822938754,851527314,857962914,870861618

%N Numbers that are the sum of four fourth powers in exactly nine ways.

%C Differs from A344926 at term 5 because 328118259 is a term because 328118259 = 2^4 + 77^4 + 109^4 + 111^4 = 8^4 + 79^4 + 93^4 + 121^4 = 18^4 + 79^4 + 97^4 + 119^4 = 21^4 + 77^4 + 98^4 + 119^4 = 27^4 + 77^4 + 94^4 + 121^4 = 34^4 + 77^4 + 89^4 + 123^4 = 46^4 + 57^4 + 103^4 + 119^4 = 49^4 + 77^4 + 77^4 + 126^4 = 61^4 + 66^4 + 77^4 + 127^4.

%H David Consiglio, Jr., <a href="/A344927/b344927.txt">Table of n, a(n) for n = 1..64</a>

%e 328118259 is a term because 328118259 = 2^4 + 77^4 + 109^4 + 111^4 = 8^4 + 79^4 + 93^4 + 121^4 = 18^4 + 79^4 + 97^4 + 119^4 = 21^4 + 77^4 + 98^4 + 119^4 = 27^4 + 77^4 + 94^4 + 121^4 = 34^4 + 77^4 + 89^4 + 123^4 = 46^4 + 57^4 + 103^4 + 119^4 = 49^4 + 77^4 + 77^4 + 126^4 = 61^4 + 66^4 + 77^4 + 127^4.

%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, 4):

%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. A341892, A344751, A344925, A344926, A344929, A345154.

%K nonn

%O 1,1

%A _David Consiglio, Jr._, Jun 02 2021