A126200 Numbers n such that n^2 is a sum of consecutive cubes larger than 1.

8, 27, 64, 125, 204, 216, 312, 315, 323, 343, 504, 512, 588, 720, 729, 1000, 1331, 1728, 2079, 2170, 2197, 2744, 2940, 3375, 4096, 4472, 4913, 4914, 5187, 5832, 5880, 5984, 6630, 6859, 7497, 8000, 8721, 8778, 9261, 9360, 10296, 10648, 10695, 11024, 12167, 13104

Offset: 1

Author

Zak Seidov, Mar 11 2007

Keywords

• nonn

Comments

Note that all triangular numbers A000217(i) have squares A000217(i)^2=A000537(i), which are sums of consecutive cubes starting with 1. But such decompositions are not counted here. - R. J. Mathar, Nov 02 2007

Also, the positive integers n such that n^2 is the difference of squares of two positive triangular numbers. - Max Alekseyev, Jul 27 2014

Included all cubes > 1.

Examples

			204^2=23^3+24^3+25^3, 312^2=14^3+15^3+...24^3+25^3;
n^2=sum[i^3, (i=i1...i2)]; {n, i1=initial index of cube, i2=final index of cube}: {8, 4, 4}, {27, 9, 9}, {64, 16, 16}, {125, 25, 25}, {204, 23, 25}, {216, 36, 36}, {312, 14, 25}, {315, 25, 29}, {323, 9, 25}, {343, 49, 49}, {504, 28, 35}, {512, 64, 64}, {588, 14, 34}, {720, 25, 39}, {729, 81, 81}, {1000, 100, 100}, {1331, 121, 121}, {1728, 144, 144}, {2079, 33, 65}, {2170, 96, 100}, {2197, 169, 169}, {2744, 196, 196}.
		

Links

• Donovan Johnson, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A126203.

Programs

PARI

mc=335241; cb=vector(mc); for(i=2, mc, cb[i]=i^3); v=vector(1000); mx=194104539^2; n=0; for(i=2, mc, s=0; for(j=i, mc, s=s+cb[j]; if(s>mx, next(2)); if(issquare(s,&sr), n++; v[n]=sr))); v=vecsort(v); for(i=1, 1000, write("b126200.txt", i " " v[i])) /* Donovan Johnson, Feb 02 2013 */

Extensions

Better definition from Dean Hickerson, Dec 02 2007

Many terms were missing - thanks to Donovan Johnson for catching this. (Feb 02 2013)