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

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

Author

Omar E. Pol, Jan 14 2015

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

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

Crossrefs

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

Partial sums of A139140.

Programs

Mathematica

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

PARI

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

Python

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

Formula

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