A103532 Number of divisors of 240^n.

1, 20, 81, 208, 425, 756, 1225, 1856, 2673, 3700, 4961, 6480, 8281, 10388, 12825, 15616, 18785, 22356, 26353, 30800, 35721, 41140, 47081, 53568, 60625, 68276, 76545, 85456, 95033, 105300, 116281, 128000, 140481, 153748, 167825, 182736

Offset: 0

Author

J. Lowell, Aug 30 2008

Keywords

• nonn
• easy

Comments

Geometric interpretation: Take a simple cubical grid of size (2n+1). Number the coordinates along each axis from 1 to (2n+1). Select only the cells that have at least two odd coordinates, and discard the rest. The number of selected cells is a(n). - Arun Giridhar, Mar 27 2015

Examples

			a(2) = 81 because 240^2 has 81 divisors.
a(2) = 81 because a 5 X 5 X 5 grid has 81 cells with at least two odd coordinates each, coordinate numbering starting at 1.
		

Links

• Kelvin Voskuijl, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. similar sequences, with the formula (k*n-k+2)*n^2/2, listed in A262000.

Programs

Magma

[(4*n+1)*(n+1)^2: n in [0..45]]; // Vincenzo Librandi, Feb 10 2016

Maple

A103532 := proc(n) (4*n+1)*(n+1)^2 ; end proc: # R. J. Mathar, Aug 31 2008

Mathematica

Table[(4 n + 1) (n + 1)^2, {n, 0, 40}] (* Stefan Steinerberger, Aug 31 2008 *)
DivisorSigma[0,240^Range[0,40]] (* or *) LinearRecurrence[{4,-6,4,-1},{1,20,81,208},40] (* Harvey P. Dale, Jan 21 2013 *)

Formula

From R. J. Mathar and Stefan Steinerberger, Aug 31 2008: (Start)

a(n) = (4*n+1)*(n+1)^2.

G.f.: (1+16x+7x^2)/(1-x)^4.

Inverse binomial transform: 1, 19, 42, 24, 0 (0 continued). (End)

a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>3. - Harvey P. Dale, Jan 21 2013

a(n) = (n+1)*A001107(n+1), where A001107 are the partial sums of A017007. - J. M. Bergot, Jul 08 2013

a(n) = Sum_{i=0..n} (n+1)*(8*i+1). [Bruno Berselli, Sep 08 2015]

Sum_{n>=0} 1/a(n) = 2*Pi/9 - Pi^2/18 + 4*log(2)/3 = 1.07401658592825... . - Vaclav Kotesovec, Oct 04 2016

E.g.f.: exp(x)*(1 + 19*x + 21*x^2 + 4*x^3). - Stefano Spezia, Jan 31 2025

Sum_{n>=0} (-1)^n/a(n) = 2*sqrt(2)*Pi/9 - Pi^2/36 - (4/9)*(log(2) + sqrt(2)*log(sqrt(2)-1)). - Amiram Eldar, Aug 15 2025

Extensions

More terms from Stefan Steinerberger and R. J. Mathar, Aug 31 2008

Example corrected by Harvey P. Dale, Jan 21 2013