A305444 a(n) = Product_{p is odd and prime and divisor of n} (p - 2).

1, 1, 1, 1, 3, 1, 5, 1, 1, 3, 9, 1, 11, 5, 3, 1, 15, 1, 17, 3, 5, 9, 21, 1, 3, 11, 1, 5, 27, 3, 29, 1, 9, 15, 15, 1, 35, 17, 11, 3, 39, 5, 41, 9, 3, 21, 45, 1, 5, 3, 15, 11, 51, 1, 27, 5, 17, 27, 57, 3, 59, 29, 5, 1, 33, 9, 65, 15, 21, 15, 69, 1, 71, 35, 3, 17

Offset: 1

Author

Markus Sigg, Aug 12 2018

Keywords

• nonn
• easy
• mult

Comments

Denominator of c_n = Product_{odd p| n} (p-1)/(p-2). Numerator is A173557. [Yamasaki and Yamasaki]. - N. J. A. Sloane, Jan 19 2020

This ratio, multiplied by the twin prime constant, occurs in the asymptotic behavior of prime gaps of size 2*n as decribed by the Hardy-Littlewood asymptotic conjecture for the number of prime pairs. See A005597 for more information. - Hugo Pfoertner, Dec 25 2024

Links

• Markus Sigg, Table of n, a(n) for n = 1..10000
• Hugo Pfoertner, Plot of A173557(n)/a(n) vs n, using Plot 2.
• Yasuo Yamasaki and Aiichi Yamasaki, On the Gap Distribution of Prime Numbers, Kyoto University Research Information Repository, October 1994. MR1370273 (97a:11141).

Crossrefs

Cf. A173557, A000010, A008683, A001221.

Programs

Maple

A305444 := proc(n) mul(d - 2, d = numtheory[factorset](n) minus {2}) end proc:

Mathematica

a[n_] := If[n == 1, 1, Times @@ (DeleteCases[FactorInteger[n][[All, 1]], 2] - 2)];
Array[a, 100] (* Jean-François Alcover, Apr 08 2020*)

PARI

a(n)={my(f=factor(n>>valuation(n,2))[,1]); prod(i=1, #f, f[i]-2)} \\ Andrew Howroyd, Aug 12 2018

Python

from math import prod
from sympy import primefactors
def A305444(n): return prod(p-2 for p in primefactors(n>>(~n&n-1).bit_length())) # Chai Wah Wu, Sep 08 2023

Formula

Sum_{k=1..n} a(k) ~ c * n^2, where c = (2/3) * Product_{p prime} (1 - 3/(p*(p+1))) = 0.1950799046... . - Amiram Eldar, Nov 12 2022

a(n) = abs( Sum_{d divides n, d odd} mobius(d) * phi(d) ). - Peter Bala, Feb 01 2024

a(n) = (-1)^omega(n) * Sum_{d|n} mu(d)*phi(2*d), where omega = A001221. - Ridouane Oudra, Jul 30 2025