A066215 - OEIS

%I #21 May 27 2024 08:06:29

%S 1,8,27,36,64,72,125,216,252,288,343,378,512,520,576,584,729,738,756,

%T 792,828,855,954,972,1000,1044,1331,1350,1440,1520,1540,1728,1764,

%U 1800,1890,1944,1980,2016,2070,2160,2197,2304,2352,2376,2400,2484,2520,2548

%N Numbers which are sums of cubes of some subset of divisors.

%C There are cubes that have not a single, trivial representation but more than one. These start with 27000 =+8^3+9^3+10^3+12^3+15^3+18^3+24^3 =+1^3+2^3+4^3+6^3+8^3+10^3+15^3+20^3+24^3 = +2^3+4^3+9^3+10^3+15^3+20^3+24^3 =+1^3+2^3+4^3+12^3+15^3+20^3+24^3 =+2^3+3^3+4^3+5^3+6^3+12^3+15^3+18^3+25^3 =+1^3+4^3+5^3+6^3+8^3+9^3+12^3+20^3+25^3 =+15^3+20^3+25^3 = +1^3+3^3+6^3+8^3+9^3+18^3+27^3 =+30^3 and 46656 =+1^3+2^3+3^3+6^3+8^3+9^3+12^3+16^3+18^3+24^3+27^3 =+4^3+24^3+32^3 =+36^3 and 74088 =+2^3+6^3+7^3+8^3+9^3+12^3+18^3+21^3+24^3+27^3+28^3 =+4^3+6^3+8^3+14^3+18^3+21^3+24^3+27^3+28^3 =+42^3. - _R. J. Mathar_, Jan 21 2024

%C If m is in the sequence then so is m*k^3 for k >= 1. - _David A. Corneth_, Jan 21 2024

%H David A. Corneth, <a href="/A066215/b066215.txt">Table of n, a(n) for n = 1..10286</a> (first 649 terms from R. J. Mathar)

%H R. J. Mathar, <a href="/A066215/a066215.txt">Examples/decompositions for entries <10000</a>

%e 72 is in the list since 72 = 2^3 + 4^3 and 2 and 4 are divisors of 72

%t okQ[k_] := AnyTrue[Subsets[Select[Divisors[k]^3, # <= k&]], Total[#]==k&];

%t Reap[For[k = 1, k <= 10000, k++, If[okQ[k], Print[k]; Sow[k]]]][[2, 1]] (* _Jean-François Alcover_, May 27 2024 *)

%Y Cf. A000578, A066214, A066213, A066216.

%K nonn

%O 1,2

%A _Erich Friedman_, Dec 17 2001