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A034665 Sum of n-th powers of divisors of 32.

%I A034665 #41 Sep 08 2022 08:44:52
%S A034665 6,63,1365,37449,1118481,34636833,1090785345,34630287489,
%T A034665 1103823438081,35253226045953,1127000493261825,36046397799139329,
%U A034665 1153203048319815681,36897992296869404673,1180663682709764194305
%N A034665 Sum of n-th powers of divisors of 32.
%H A034665 T. D. Noe, <a href="/A034665/b034665.txt">Table of n, a(n) for n = 0..200</a>
%H A034665 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 A034665 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (63,-1302,11160,-41664,64512,-32768).
%F A034665 G.f.: -3*(21504*x^5-27776*x^4+11160*x^3-1736*x^2+105*x-2) / ((x-1)*(2*x-1)*(4*x-1)*(8*x-1)*(16*x-1)*(32*x-1)). - _Colin Barker_, Apr 20 2014
%F A034665 a(n) = (2^(6*n) - 1)/( 2^n - 1). Exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 63*x + 2667*x^2 + 97155*x^3 + ... is the o.g.f. for the 5th subdiagonal of triangle A022166, essentially A006110. - _Peter Bala_, Apr 07 2015
%F A034665 a(n) = 1 + 2^n + 4^n + 8^n + 16^n + 32^n for n>=0. - _Karl V. Keller, Jr._, Feb 02 2021
%t A034665 Total[#^Range[0, 15]&/@Divisors[32]] (* _Vincenzo Librandi_, Apr 17 2014 *)
%t A034665 LinearRecurrence[{63,-1302,11160,-41664,64512,-32768},{6,63,1365,37449,1118481,34636833},20] (* _Harvey P. Dale_, Jan 10 2015 *)
%o A034665 (Sage) [sigma(32,n)for n in range(0,15)] # _Zerinvary Lajos_, Jun 04 2009
%o A034665 (Magma) [DivisorSigma(n,32): n in [0..15]]; // _Vincenzo Librandi_, Apr 17 2014
%o A034665 (PARI) a(n)=(64^n-1)/(2^n-1) \\ _Charles R Greathouse IV_, Oct 07 2015
%o A034665 (Python) print([1+2**n+4**n+8**n+16**n+32**n for n in range(15)]) # _Karl V. Keller, Jr._, Feb 02 2021
%Y A034665 Cf. A006110, A022166.
%K A034665 nonn,easy
%O A034665 0,1
%A A034665 _N. J. A. Sloane_