%I #37 Sep 08 2022 08:46:08

%S 1,6,34,307,3433,50883,874427,17717436,405157609,10414924259,

%T 295726594871,9214021138217,312089127730471,11424774176377721,

%U 449318695089164129,18896344248070459234,846136606134407223412,40192694877626586149007,2018612350537940175272987

%N a(n) = sigma_1(1) + sigma_2(2) + sigma_3(3) + ... + sigma_n-1(n-1) + sigma_n(n).

%C Let sigma_k(n) represent the sum of the k-th powers of the divisors of n.

%C Then a(n) = Sum_{k=1..n} sigma_k(k), the partial sums of sigma_k(k) for k from 1 to n.

%C Partial sums of A023887.

%H Jens Kruse Andersen, <a href="/A245466/b245466.txt">Table of n, a(n) for n = 1..100</a>

%F a(n) = Sum_{k=1..n} sigma_k(k).

%F a(1) = 1. a(n) = a(n-1) + sigma_n(n), for n > 1. - _Jens Kruse Andersen_, Jul 29 2014

%F a(n) = n + Sum_{d=2..n} (d^(d*(floor(n/d)+1))-d^d)/(d^d-1). - _Chayim Lowen_, Aug 04 2015

%e a(1) = 1 because sigma_1(1) = sigma(1) = 1.

%e a(2) = 6: sigma_1(1) + sigma_2(2) = 1 + (1^2 + 2^2) = 6.

%e a(3) = 34: sigma_1(1) + sigma_2(2) + sigma_3(3) = 6 + (1^3 + 3^3) = 34.

%e a(4) = 307: sigma_1(1) + ... + sigma_4(4) = 34 + (1^4 + 2^4 + 4^4) = 307.

%p B:= [seq(numtheory:-sigma[n](n),n=1..100)]:

%p seq(add(B[i],i=1..n),n=1..100); # _Robert Israel_, Jul 28 2014

%t Table[Sum[DivisorSigma[k, k], {k, n}], {n, 20}]

%t Accumulate[Table[DivisorSigma[n,n],{n,20}]] (* _Harvey P. Dale_, Apr 10 2018 *)

%o (PARI)

%o a(n) = sum(i=1,n,sigma(i,i))

%o vector(50, n, a(n)) \\ _Derek Orr_, Jul 27 2014

%o (Magma) [&+[DivisorSigma(i, i): i in [1..n]]: n in [1..20]]; // _Bruno Berselli_, Jul 29 2014

%o (Magma) [n eq 1 select 1 else Self(n-1)+ DivisorSigma(n, n): n in [1..20]]; // _Vincenzo Librandi_, Aug 05 2015

%Y Cf. A000203, A001157-A001160, A013954-A013958, A023887.

%K nonn

%O 1,2

%A _Wesley Ivan Hurt_, Jul 22 2014