A244583 a(n) = sum of all divisors of all positive integers <= prime(n).

4, 8, 21, 41, 99, 141, 238, 297, 431, 690, 794, 1136, 1384, 1524, 1806, 2304, 2846, 3076, 3699, 4137, 4406, 5128, 5645, 6499, 7755, 8401, 8721, 9393, 9783, 10513, 13280, 14095, 15443, 15871, 18232, 18756, 20320, 21873, 22875, 24604, 26274, 27002, 29982, 30684

Offset: 1

Author

Omar E. Pol, Jun 30 2014

Keywords

• nonn

Comments

Limit_{n->oo} a(n)/prime(n)^2 = zeta(2)/2 = Pi^2/12 = A072691 = 0.82246703342.... For example, at n = 2*10^6, the ratio converges to 0.822467033... (+-2 in the last digit with increments on n of +100). If the ratio is calculated with a nonprime for the upper summation limit then the ratio runs slightly larger and converges slower. See formula section of A024916 for the general case. - Richard R. Forberg, Jan 04 2015

This is a subsequence of A024916 therefore a(n) also has a symmetric representation. For more information see A236104, A237593. - Omar E. Pol, Jan 05 2015

Crossrefs

Cf. A000040, A000203, A024916, A236104, A237593, A237270, A237271, A244576, A244578.

Partial sums of A358683.

Programs

Mathematica

a244583[n_] := Sum[DivisorSigma[1, i], {i, #}] & /@ Prime[Range@n]; a244583[44] (* Michael De Vlieger, Jan 06 2015 *)

PARI

a(n) = sum(i=1, prime(n), sigma(i)); \\ Michel Marcus, Sep 29 2014

Python

from math import isqrt
from sympy import prime
def A244583(n): return -(s:=isqrt(p:=prime(n)))**2*(s+1) + sum((q:=p//k)*((k<<1)+q+1) for k in range(1,s+1))>>1 # Chai Wah Wu, Oct 23 2023

Formula

a(n) = A024916(A000040(n)).

a(n) = A001248(n) - A050482(n). - Omar E. Pol, Jan 05 2015

Extensions

More terms from Michel Marcus, Sep 29 2014