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

%I A305444 #85 Aug 01 2025 05:51:23
%S A305444 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 A305444 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 A305444 57,3,59,29,5,1,33,9,65,15,21,15,69,1,71,35,3,17
%N A305444 a(n) = Product_{p is odd and prime and divisor of n} (p - 2).
%C A305444 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
%C A305444 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
%H A305444 Markus Sigg, <a href="/A305444/b305444.txt">Table of n, a(n) for n = 1..10000</a>
%H A305444 Hugo Pfoertner, <a href="/plot2a?name1=A173557&amp;name2=A305444&amp;tform1=untransformed&amp;tform2=untransformed&amp;shift=0&amp;radiop1=ratio&amp;drawpoints=true">Plot of A173557(n)/a(n) vs n</a>, using Plot 2.
%H A305444 Yasuo Yamasaki and Aiichi Yamasaki, <a href="https://web.archive.org/web/20230423052037/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 A305444 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 A305444 a(n) = abs( Sum_{d divides n, d odd} mobius(d) * phi(d) ). - _Peter Bala_, Feb 01 2024
%F A305444 a(n) = (-1)^omega(n) * Sum_{d|n} mu(d)*phi(2*d), where omega = A001221. - _Ridouane Oudra_, Jul 30 2025
%p A305444 A305444 := proc(n) mul(d - 2, d = numtheory[factorset](n) minus {2}) end proc:
%t A305444 a[n_] := If[n == 1, 1, Times @@ (DeleteCases[FactorInteger[n][[All, 1]], 2] - 2)];
%t A305444 Array[a, 100] (* _Jean-François Alcover_, Apr 08 2020*)
%o A305444 (PARI) a(n)={my(f=factor(n>>valuation(n,2))[,1]); prod(i=1, #f, f[i]-2)} \\ _Andrew Howroyd_, Aug 12 2018
%o A305444 (Python)
%o A305444 from math import prod
%o A305444 from sympy import primefactors
%o A305444 def A305444(n): return prod(p-2 for p in primefactors(n>>(~n&n-1).bit_length())) # _Chai Wah Wu_, Sep 08 2023
%Y A305444 Cf. A173557, A000010, A008683, A001221.
%K A305444 nonn,easy,mult
%O A305444 1,5
%A A305444 _Markus Sigg_, Aug 12 2018