A132106 a(n) = 1 + floor(sqrt(n)) + Sum_{i=1..n} floor(n/i).

1, 3, 5, 7, 11, 13, 17, 19, 23, 27, 31, 33, 39, 41, 45, 49, 55, 57, 63, 65, 71, 75, 79, 81, 89, 93, 97, 101, 107, 109, 117, 119, 125, 129, 133, 137, 147, 149, 153, 157, 165, 167, 175, 177, 183, 189, 193, 195, 205, 209, 215, 219, 225, 227, 235, 239, 247, 251, 255, 257, 269

Offset: 0

Author

Richard C. Schroeppel, Oct 31 2007

Keywords

• nonn

Comments

Matches first 8 odd primes.

Links

• Antti Karttunen, Table of n, a(n) for n = 0..16384
• Aidan Botkin, Madeline L. Dawsey, David J. Hemmer, Matthew R. Just, and Robert Schneider, Partition-theoretic model of prime distribution, arXiv:2501.00580 [math.NT], 2024. See page 6.

Crossrefs

Cf. A000196, A006218, A161841 (first differences).

Programs

Mathematica

a[n_]:=1+2Sum[Ceiling[DivisorSigma[0,k]/2],{k,n-1}]; Array[a,61] (* Stefano Spezia, Jan 06 2025 *)

PARI

a(n) = 1 + sqrtint(n) + sum(i=1, n, n\i); \\ Michel Marcus, Jun 17 2021

Python

from math import isqrt
def A132106(n): return (lambda m: 2*(sum(n//k for k in range(1, m+1)))+m*(1-m)+1)(isqrt(n)) # Chai Wah Wu, Oct 08 2021

Formula

a(n) = 1 + Sum_{i=1..n} A161841(i) for n > 0. - Christian Krause, Jun 17 2021

a(n) = 1 + A000196(n) + A006218(n). - Vaclav Kotesovec, Jul 01 2021