A244576 -id:A244576 - cp's OEIS frontend

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

Omar E. Pol, Jun 30 2014

nonn

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

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

Partial sums of A358683.

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

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

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

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

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

More terms from Michel Marcus, Sep 29 2014

A244578 Sum of all aliquot divisors of all positive integers <= prime(n).

Original entry on oeis.org

1, 2, 6, 13, 33, 50, 85, 107, 155, 255, 298, 433, 523, 578, 678, 873, 1076, 1185, 1421, 1581, 1705, 1968, 2159, 2494, 3002, 3250, 3365, 3615, 3788, 4072, 5152, 5449, 5990, 6141, 7057, 7280, 7917, 8507, 8847, 9553, 10164, 10531, 11646, 11963, 12408, 12679

Offset: 1

Omar E. Pol, Jun 30 2014

nonn

Cf. A000040, A000203, A001065, A153485, A244576, A244583.

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

a(n) = A153485(A000040(n)).

a(n) ~ (Pi^2/12 - 1/2) * n^2 * log(n)^2. - Amiram Eldar, Mar 22 2024

More terms from Michel Marcus, Sep 29 2014

A253769 Sum of number of divisors of all positive integers <= prime(n).

Original entry on oeis.org

3, 5, 10, 16, 29, 37, 52, 60, 76, 103, 113, 142, 160, 170, 188, 219, 249, 263, 294, 314, 328, 358, 379, 413, 461, 484, 494, 516, 530, 554, 637, 659, 697, 707, 768, 782, 822, 858, 878, 919, 953, 973, 1033, 1049, 1072, 1086, 1168, 1240, 1267, 1281, 1307, 1343, 1365, 1423, 1468, 1504, 1544, 1562, 1604, 1632, 1642, 1709

Offset: 1

Omar E. Pol, Jan 14 2015

nonn

a(n) is the index of the first position of prime(n) in A027750, the sequence that lists the divisors of all integers. - Michel Marcus, Oct 17 2015

			For n = 3 the third prime number is 5 and the sum of the number of divisors of the first five positive integers is 1 + 2 + 2 + 3 + 2 = 10, so a(3) = 10.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A000005, A000040, A006218, A027750, A244576, A244578, A244583.

Partial sums of A139140.

Module[{nn=300,d},d=Accumulate[DivisorSigma[0,Range[nn]]];Table[d[[k]],{k,Prime[Range[PrimePi[nn]]]}]] (* Harvey P. Dale, Jul 12 2025 *)

a(n) = sum(i=1, prime(n), numdiv(i)); \\ Michel Marcus, Jan 15 2015

from math import isqrt
from sympy import prime
def A253769(n): return (lambda m, p: 2*sum(p//k for k in range(1, m+1))-m*m)(isqrt(prime(n)),prime(n)) # Chai Wah Wu, Oct 09 2021

a(n) = A006218(A000040(n)).

cp's OEIS Frontend

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

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A244578 Sum of all aliquot divisors of all positive integers <= prime(n).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

Extensions

A253769 Sum of number of divisors of all positive integers <= prime(n).

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Author

Keywords

Comments

Examples

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Crossrefs

Programs

Mathematica

PARI

Python

Formula