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

1,5

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

Links

Markus Sigg, Table of n, a(n) for n = 1..10000

Yasuo Yamasaki and Aiichi Yamasaki, On the Gap Distribution of Prime Numbers, Kyoto University Research Information Repository, October 1994. MR1370273 (97a:11141).

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

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*)

Prog

(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

Crossrefs

Cf. A173557.

Sequence in context: A339903 A187367 A307410 * A002945 A171232 A093423

Adjacent sequences: A305441 A305442 A305443 * A305445 A305446 A305447

Keyword

nonn,easy,mult

Author

Markus Sigg, Aug 12 2018

Status

approved