A164577 Integer averages of the first perfect cubes up to some n^3.

1, 12, 25, 45, 112, 162, 225, 396, 507, 637, 960, 1156, 1377, 1900, 2205, 2541, 3312, 3750, 4225, 5292, 5887, 6525, 7936, 8712, 9537, 11340, 12321, 13357, 15600, 16810, 18081, 20812, 22275, 23805, 27072, 28812, 30625, 34476, 36517, 38637, 43120

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Aug 16 2009

Keywords

• nonn
• easy

Comments

Integers of the form A000537(k)/k, created by the k>0 listed in A042965. - R. J. Mathar, Aug 20 2009

Integers of the form (1/4)*n*(n+1)^2 for some n. - Zak Seidov, Aug 17 2009

Examples

			The average of the first cube is 1^3/1=1=a(1).
The average of the first two cubes is (1^3+2^3)/2=9/2, not integer, and does not contribute to the sequence.
The average of the first three cubes is (1^3+2^3+3^3)/3=12, integer, and defines a(2).
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,3,-3,0,-3,3,0,1,-1).

Crossrefs

Cf. A000537, A050248, A051456, A078617, A078618, A154293, A164576

Programs

Mathematica

Timing[s=0;lst={};Do[a=(s+=n^3)/n;If[Mod[a,1]==0,AppendTo[lst,a]],{n, 5!}];lst]
With[{nn=80},Select[#[[1]]/#[[2]]&/@Thread[{Accumulate[Range[ nn]^3],Range[ nn]}],IntegerQ]] (* or *) LinearRecurrence[{1,0,3,-3,0,-3,3,0,1,-1},{1,12,25,45,112,162,225,396,507,637},50] (* Harvey P. Dale, Mar 14 2020 *)

Formula

G.f.: ( x*(1+11*x+13*x^2+17*x^3+34*x^4+11*x^5+6*x^6+3*x^7) ) / ( (1+x+x^2)^3*(x-1)^4 ). - R. J. Mathar, Jan 25 2011

Extensions

Changed comments to examples - R. J. Mathar, Aug 20 2009