This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A307410 #25 Mar 03 2025 13:32:57
%S A307410 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 A307410 1,9,15,5,1,35,17,11,3,39,5,41,9,3,21,45,1,5,3,15,11,51,1,27,5,17,27,
%U A307410 57,3,59,29,5,1,11,9,65,15,21,5,69,1,71,35,3,17,3,11,77,3,1,39,81,5,45
%N A307410 Numerators of the product in the singular series.
%C A307410 Differs from A305444 at n = 35, 65, 70, ...
%H A307410 Amiram Eldar, <a href="/A307410/b307410.txt">Table of n, a(n) for n = 1..10000</a>
%H A307410 John Omielan, <a href="https://math.stackexchange.com/a/3178061/8530">How do you compute the singular series?</a>, Mathematics Stack Exchange.
%H A307410 Terence Tao, <a href="https://terrytao.wordpress.com/2017/07/06/correlations-of-the-von-mangoldt-and-higher-divisor-functions-i-long-shift-ranges/">Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges</a>. See the next formula after equation 2.
%F A307410 a(n) = numerator of Product_{p|n;p>2}(p-2)/(p-1) where p is a prime number.
%F A307410 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A380839(k) = 2 * Product_{p prime} (1-1/(p^2-p)) = 2 * A005596 = 0.7479116272384045761094... . - _Amiram Eldar_, Mar 03 2025
%p A307410 f:= proc(n) numer(mul((p-2)/(p-1),p=select(type,numtheory:-factorset(n),odd))) end proc:
%p A307410 map(f, [$1..100]); # _Robert Israel_, Apr 07 2019
%t A307410 Table[Times@@(DeleteDuplicates[DeleteCases[DeleteCases[Exp[MangoldtLambda[Divisors[h]]], 1],2]] - 2)/Times@@(DeleteDuplicates[DeleteCases[DeleteCases[Exp[MangoldtLambda[Divisors[h]]], 1], 2]] - 1), {h, 1, 85}]
%t A307410 Numerator[%]
%t A307410 f[p_, e_] := If[p == 2, 1, (p-2)/(p-1)]; a[n_] := Numerator[Times @@ f @@@ FactorInteger[n]]; a[1] = 1; Array[a, 100] (* _Amiram Eldar_, Mar 03 2025 *)
%o A307410 (PARI) a(n) = my(f=factor(n)[,1]~); numerator(prod(k=1, #f, if (f[k]>2, (f[k]-2)/(f[k]-1), 1))); \\ _Michel Marcus_, Apr 07 2019
%Y A307410 Cf. A005596, A005597, A305444, A380839 (denominators).
%K A307410 nonn,frac,look
%O A307410 1,5
%A A307410 _Mats Granvik_, Apr 07 2019