A245466 a(n) = sigma_1(1) + sigma_2(2) + sigma_3(3) + ... + sigma_n-1(n-1) + sigma_n(n).

1, 6, 34, 307, 3433, 50883, 874427, 17717436, 405157609, 10414924259, 295726594871, 9214021138217, 312089127730471, 11424774176377721, 449318695089164129, 18896344248070459234, 846136606134407223412, 40192694877626586149007, 2018612350537940175272987

Offset: 1

Author

Wesley Ivan Hurt, Jul 22 2014

Keywords

• nonn

Comments

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

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

Partial sums of A023887.

Examples

			a(1) = 1 because sigma_1(1) = sigma(1) = 1.
a(2) = 6: sigma_1(1) + sigma_2(2) = 1 + (1^2 + 2^2) = 6.
a(3) = 34: sigma_1(1) + sigma_2(2) + sigma_3(3) = 6 + (1^3 + 3^3) = 34.
a(4) = 307: sigma_1(1) + ... + sigma_4(4) = 34 + (1^4 + 2^4 + 4^4) = 307.
		

Links

• Jens Kruse Andersen, Table of n, a(n) for n = 1..100

Crossrefs

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

Programs

Magma

[&+[DivisorSigma(i, i): i in [1..n]]: n in [1..20]]; // Bruno Berselli, Jul 29 2014

Magma

[n eq 1 select 1 else Self(n-1)+ DivisorSigma(n, n): n in [1..20]]; // Vincenzo Librandi, Aug 05 2015

Maple

B:= [seq(numtheory:-sigma[n](n),n=1..100)]:
seq(add(B[i],i=1..n),n=1..100); # Robert Israel, Jul 28 2014

Mathematica

Table[Sum[DivisorSigma[k, k], {k, n}], {n, 20}]
Accumulate[Table[DivisorSigma[n,n],{n,20}]] (* Harvey P. Dale, Apr 10 2018 *)

PARI

a(n) = sum(i=1,n,sigma(i,i))
vector(50, n, a(n)) \\ Derek Orr, Jul 27 2014

Formula

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

a(1) = 1. a(n) = a(n-1) + sigma_n(n), for n > 1. - Jens Kruse Andersen, Jul 29 2014

a(n) = n + Sum_{d=2..n} (d^(d*(floor(n/d)+1))-d^d)/(d^d-1). - Chayim Lowen, Aug 04 2015