A307410 Numerators of the product in the singular series.

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, 5, 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, 11, 9, 65, 15, 21, 5, 69, 1, 71, 35, 3, 17, 3, 11, 77, 3, 1, 39, 81, 5, 45

Offset: 1

Author

Mats Granvik, Apr 07 2019

Keywords

• nonn
• frac
• look

Comments

Differs from A305444 at n = 35, 65, 70, ...

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• John Omielan, How do you compute the singular series?, Mathematics Stack Exchange.
• Terence Tao, Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges. See the next formula after equation 2.

Crossrefs

Cf. A005596, A005597, A305444, A380839 (denominators).

Programs

Maple

f:= proc(n) numer(mul((p-2)/(p-1),p=select(type,numtheory:-factorset(n),odd))) end proc:
map(f, [$1..100]); # Robert Israel, Apr 07 2019

Mathematica

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}]
Numerator[%]
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 *)

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

Formula

a(n) = numerator of Product_{p|n;p>2}(p-2)/(p-1) where p is a prime number.

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