%I #39 Oct 26 2025 11:33:49

%S 4,15,85,585,4369,33825,266305,2113665,16843009,134480385,1074791425,

%T 8594130945,68736258049,549822930945,4398314962945,35185445863425,

%U 281479271743489,2251816993685505,18014467229220865,144115462954287105,1152922604119523329,9223376434903384065

%N Sum of n-th powers of divisors of 8.

%C Conjecture: No primes in this sequence (checked for first 10000 terms). - _Artur Jasinski_, Sep 23 2008

%C All terms are composite because a(n) = (1 + 2^n)*(1 + 4^n). - _T. D. Noe_, Apr 26 2010

%H T. D. Noe, <a href="/A034496/b034496.txt">Table of n, a(n) for n = 0..200</a>

%H Quynh Nguyen, Jean Pedersen, and Hien T. Vu, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Pedersen/pedersen2.html">New Integer Sequences Arising From 3-Period Folding Numbers</a>, Vol. 19 (2016), Article 16.3.1. See Table 1.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (15,-70,120,-64).

%F 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]

%F From _Peter Bala_, Apr 07 2015: (Start)

%F a(n) = (2^(4*n) - 1)/( 2^n - 1) = 1 + 2^n + 4^n + 8^n.

%F 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. (End)

%t Total[#^Range[0, 20]&/@Divisors[8]] (* _Vincenzo Librandi_, Apr 17 2014 *)

%t DivisorSigma[Range[0,20],8] (* _Harvey P. Dale_, May 16 2020 *)

%o (SageMath) [sigma(8,n) for n in range(0,19)] # _Zerinvary Lajos_, Jun 04 2009

%o (PARI) a(n)=sigma(8, n) \\ _Charles R Greathouse IV_, May 16 2011

%o (Magma) [DivisorSigma(n,8): n in [0..20]]; // _Vincenzo Librandi_, Apr 17 2014

%Y Cf. A006096, A022166.

%K nonn,easy

%O 0,1

%A _N. J. A. Sloane_