|
%I #75 Feb 03 2024 10:15:44
%S 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,
%T 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,
%U 57,3,59,29,5,1,33,9,65,15,21,15,69,1,71,35,3,17
%N a(n) = Product_{p is odd and prime and divisor of n} (p - 2).
%C 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
%H Markus Sigg, <a href="/A305444/b305444.txt">Table of n, a(n) for n = 1..10000</a>
%H Yasuo Yamasaki and Aiichi Yamasaki, <a href="https://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/84326/1/0887-10.pdf">On the Gap Distribution of Prime Numbers</a>, Kyoto University Research Information Repository, October 1994. MR1370273 (97a:11141).
%F 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
%F a(n) = abs( Sum_{d divides n, d odd} mobius(d) * phi(d) ). - _Peter Bala_, Feb 01 2024
%p A305444 := proc(n) mul(d - 2, d = numtheory[factorset](n) minus {2}) end proc:
%t a[n_] := If[n == 1, 1, Times @@ (DeleteCases[FactorInteger[n][[All, 1]], 2] - 2)];
%t Array[a, 100] (* _Jean-François Alcover_, Apr 08 2020*)
%o (PARI) a(n)={my(f=factor(n>>valuation(n,2))[,1]); prod(i=1, #f, f[i]-2)} \\ _Andrew Howroyd_, Aug 12 2018
%o (Python)
%o from math import prod
%o from sympy import primefactors
%o def A305444(n): return prod(p-2 for p in primefactors(n>>(~n&n-1).bit_length())) # _Chai Wah Wu_, Sep 08 2023
%Y Cf. A173557.
%K nonn,easy,mult
%O 1,5
%A _Markus Sigg_, Aug 12 2018
|
