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A123294 Sum of 13 positive 5th powers.

%I A123294 #13 Feb 16 2025 08:33:02
%S A123294 13,44,75,106,137,168,199,230,255,261,286,292,317,323,348,354,379,385,
%T A123294 410,416,441,472,497,503,528,534,559,565,590,596,621,627,652,683,714,
%U A123294 739,745,770,776,801,807,832,838,863,894,925,956,981,987,1012,1018,1036
%N A123294 Sum of 13 positive 5th powers.
%C A123294 Up to 416 = 13*(2^5) this sequence is identical to x+1 for x in A003357 Sum of 12 positive 5th powers. Primes in this sequence (13, 137, 199, 317, ...) are A123299. As proved by J.-R. Chen in 1964, g(5) = 37, so every positive integer can be written as the sum of no more than 37 positive 5th powers. G(5) <= 17, bounding the least integer G(5) such that every positive integer beyond a certain point (i.e., all but a finite number) is the sum of G(5) 5th powers.
%H A123294 Giovanni Resta, <a href="/A123294/b123294.txt">Table of n, a(n) for n = 1..10000</a>
%H A123294 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WaringsProblem.html">Waring's Problem</a>.
%F A123294 Sumset {A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584}.
%e A123294 a(1) = 13 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5.
%e A123294 a(2) = 44 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5.
%e A123294 a(9) = 255 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 3^5.
%e A123294 a(11) = 286 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 3^5
%t A123294 up = 1500; q = Range[up^(1/5)]^5; a = {0}; Do[b = Select[ Union@ Flatten@ Table[e + a, {e, q}], # <= up &]; a = b, {k, 13}]; a (* _Giovanni Resta_, Jun 12 2016 *)
%Y A123294 Cf. A000584, A003336, A003347, A003349, A003350, A003351, A003352, A003353, A003354, A003355, A003356, A003357, A123295-A123297, A008480, A123299.
%K A123294 easy,nonn
%O A123294 1,1
%A A123294 _Jonathan Vos Post_, Sep 24 2006
%E A123294 Two missing terms and more terms from _Giovanni Resta_, Jun 12 2016