A139140 -id:A139140 - cp's OEIS frontend

A139141 For n>=1, a(n) = floor((d(p(n)+1) + d(p(n)+2) + d(p(n)+3) + ... +d(p(n+1)))/(p(n+1) - p(n))), where d(m) is the number of positive divisors of m and p(n) is the n-th prime. a(0) = floor((d(1) + d(2))/2).

1, 2, 2, 3, 3, 4, 3, 4, 4, 4, 5, 4, 4, 5, 4, 5, 5, 7, 5, 5, 7, 5, 5, 5, 6, 5, 5, 5, 7, 6, 5, 5, 6, 5, 6, 7, 6, 6, 5, 6, 5, 10, 6, 8, 5, 7, 6, 6, 6, 7, 6, 6, 11, 5, 7, 6, 6, 9, 7, 7, 5, 6, 7, 6, 9, 6, 7, 7, 6, 7, 8, 5, 7, 7, 7, 5, 7, 8, 7, 7, 6, 13, 6, 11, 6, 8, 6, 7, 6, 9, 6, 7, 8, 6, 7, 5, 8, 7, 7, 7, 7, 6, 8

Offset: 0

Leroy Quet, Apr 10 2008

nonn

The sequence approximates the average number of divisors over all integers between consecutive primes.

			The 9th prime is 23 and the 10th prime is 29. So a(9) = floor((d(24) + d(25) + d(26) + d(27) + d(28) + d(29))/6) = floor((8 + 3 + 4 + 4 + 6 + 2)/6) = floor(27/6) = 4.

Cf. A001223, A139140.

A139141 := proc(n) if n = 0 then 1; else add(numtheory[tau](k),k=ithprime(n)+1..ithprime(n+1)) ; floor(%/(ithprime(n+1)-ithprime(n))) ; fi; end proc: seq(A139141(n),n=0..120) ; # R. J. Mathar, Oct 24 2009

For n>= 1, a(n) = floor(A139140(n)/A001223(n)).

Extended beyond a(11) by R. J. Mathar, Oct 24 2009

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

A139141 For n>=1, a(n) = floor((d(p(n)+1) + d(p(n)+2) + d(p(n)+3) + ... +d(p(n+1)))/(p(n+1) - p(n))), where d(m) is the number of positive divisors of m and p(n) is the n-th prime. a(0) = floor((d(1) + d(2))/2).

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

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