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

1,5

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.

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A176801 A339903 A187367 * A305444 A002945 A171232

Adjacent sequences: A307407 A307408 A307409 * A307411 A307412 A307413

Keyword

nonn,frac,look

Author

Mats Granvik, Apr 07 2019

Status

approved