A034496 Sum of n-th powers of divisors of 8.

4, 15, 85, 585, 4369, 33825, 266305, 2113665, 16843009, 134480385, 1074791425, 8594130945, 68736258049, 549822930945, 4398314962945, 35185445863425, 281479271743489, 2251816993685505, 18014467229220865

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Conjecture: No primes in this sequence (checked for first 10000 terms). [Artur Jasinski, Sep 23 2008]

All terms are composite because a(n) = (1 + 2^n)*(1 + 4^n). [T. D. Noe, Apr 26 2010]

Links

• T. D. Noe, Table of n, a(n) for n = 0..200
• Quynh Nguyen, Jean Pedersen, and Hien T. Vu, New Integer Sequences Arising From 3-Period Folding Numbers, Vol. 19 (2016), Article 16.3.1. See Table 1.

Crossrefs

Cf. A006096, A022166.

Programs

Magma

[DivisorSigma(n,8): n in [0..20]]; // Vincenzo Librandi, Apr 17 2014

Mathematica

Total[#^Range[0, 20]&/@Divisors[8]] (* Vincenzo Librandi, Apr 17 2014 *)
DivisorSigma[Range[0,20],8] (* Harvey P. Dale, May 16 2020 *)

PARI

a(n)=sigma(8, n) \\ Charles R Greathouse IV, May 16 2011

Sage

[sigma(8,n) for n in range(0,19)] # Zerinvary Lajos, Jun 04 2009

Formula

G.f.: (4 - 45*x + 140*x^2 - 120*x^3)/((1 - 8*x)*(1 - 4*x)*(1 - 2*x)*(1 - x)). [Bruno Berselli, Apr 17 2014]

a(n) = (2^(4*n) - 1)/( 2^n - 1) = 1 + 2^n + 4^n + 8^n. Exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 15*x + 155*x^2 + 1395*x^3 + ... is the o.g.f. for the 3rd subdiagonal of triangle A022166, essentially A006096. - Peter Bala, Apr 07 2015