A304408 If n = Product (p_j^k_j) then a(n) = Product ((p_j - 1)*(k_j + 1)).

1, 2, 4, 3, 8, 8, 12, 4, 6, 16, 20, 12, 24, 24, 32, 5, 32, 12, 36, 24, 48, 40, 44, 16, 12, 48, 8, 36, 56, 64, 60, 6, 80, 64, 96, 18, 72, 72, 96, 32, 80, 96, 84, 60, 48, 88, 92, 20, 18, 24, 128, 72, 104, 16, 160, 48, 144, 112, 116, 96, 120, 120, 72, 7, 192, 160, 132, 96, 176, 192

Offset

1,2

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Ilya Gutkovskiy, Scatter plot of a(n) up to n=50000

Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Index entries for sequences computed from indices in prime factorization

Index entries for sequences computed from exponents in factorization of n

Formula

a(n) = A000005(n)*abs(A023900(n)) = A000005(n)*A173557(n) = A000005(n)*A000010(A007947(n)).

a(p^k) = (p - 1)*(k + 1) where p is a prime and k > 0.

a(n) = 2^omega(n)*phi(n) if n is a squarefree (A005117), where omega() = A001221 and phi() = A000010.

From Vaclav Kotesovec, Nov 12 2025: (Start)

Sum_{k=1..n} a(k) ~ ((2*log(n) + 4*gamma - 1)*c + 2*d) * n^2/4, where

c = Product_{p prime} (1 - (5*p^2 + p - 2) / (p^2*(p+1)^2)) = 0.20758836569019514329433091663143001768944005...,

d = c * Sum_{p prime} 2*(4*p^3 + 2*p^2 - 3*p - 2) * log(p) / ((p-1)*(p+1)*(p^3 + 3*p^2 - p - 2)) = c * 2.85393173508558785972941925618814684884850264644...

and gamma is the Euler-Mascheroni constant A001620. (End)

Example

a(20) = a(2^2*5) = (2 - 1)*(2 + 1) * (5 - 1)*(1 + 1) = 24.

Maple

a:= n-> mul((i[1]-1)*(i[2]+1), i=ifactors(n)[2]):
seq(a(n), n=1..80);  # Alois P. Heinz, Jan 05 2021

Mathematica

a[n_] := Times @@ ((#[[1]] - 1) (#[[2]] + 1) & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 70}]
Table[DivisorSigma[0, n] EulerPhi[Last[Select[Divisors[n], SquareFreeQ]]], {n, 70}]

Prog

(PARI) a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i, 1], e=f[i, 2]); (p-1)*(e+1))} \\ Andrew Howroyd, Jul 24 2018

Crossrefs

Cf. A000005, A000010, A000026, A000040, A000302 (numbers n such that a(n) is odd), A001221, A006093, A007947, A023900, A034444, A059975, A062355, A173557, A304407, A304409, A304411, A304412.

Sequence in context: A188843 A209406 A188706 * A048767 A269851 A284457

Adjacent sequences: A304405 A304406 A304407 * A304409 A304410 A304411

Keyword

nonn,mult

Author

Ilya Gutkovskiy, May 12 2018

Status

approved