A349394 a(p^e) = p^(e-1) for prime powers, a(n) = 0 for all other n; Dirichlet convolution of A003415 (arithmetic derivative of n) with A055615 (Dirichlet inverse of n).

0, 1, 1, 2, 1, 0, 1, 4, 3, 0, 1, 0, 1, 0, 0, 8, 1, 0, 1, 0, 0, 0, 1, 0, 5, 0, 9, 0, 1, 0, 1, 16, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 7, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 32, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 27, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0

Offset

1,4

Comments

Dirichlet convolution of this sequence with Euler phi (A000010) is A300251.

Convolving this sequence with sigma (A000203) produces A319684.

With a(1) = 1 instead of 0, this would be the Dirichlet convolution of A129283 (A003415(n)+n) with A055615. Thus when we subtract A063524 from that convolution, we get this sequence. (See also A349434). Compare also to the convolution of A069359 (sequence agreeing with A003415 on squarefree numbers) with A055615, which is the characteristic function of primes, A010051. - Antti Karttunen, Nov 20 2021

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

P. Haukkanen, J. K. Merikoski and T. Tossavainen, Asymptotics of partial sums of the Dirichlet series of the arithmetic derivative, Mathematical Communications 25 (2020), 107-115.

Formula

a(n) = Sum_{d|n} A003415(n/d) * A055615(d).

a(n) = 0 unless n is a prime power (A246655), in which case a(p^e) = p^(e-1). - Sebastian Karlsson, Nov 19 2021

a(n) = A003557(n) * A069513(n). [From above] - Antti Karttunen, Nov 20 2021

Dirichlet g.f.: Sum_{p prime} 1/(p^s-p) [Follows from the D.g.f. of A003415 proved by Haukkanen et al.]. - Sebastian Karlsson, Nov 25 2021

Sum_{k=1..n} a(k) has an average value c*n, where c = A137245 = Sum_{primes p} 1/(p*log(p)) = 1.63661632335... - Vaclav Kotesovec, Mar 03 2023

Mathematica

f[p_, e_] := e/p; d[1] = 0; d[n_] := n * Plus @@ f @@@ FactorInteger[n]; a[n_] := DivisorSum[n, # * MoebiusMu[#] * d[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 19 2021 *)

Prog

(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A055615(n) = (n*moebius(n));
A349394(n) = sumdiv(n, d, A003415(n/d)*A055615(d));
(PARI) A349394(n) = { my(p=0, e); if((e=isprimepower(n, &p)), p^(e-1), 0); }; \\ (After Sebastian Karlsson's new formula) - Antti Karttunen, Nov 20 2021
(Haskell)
import Math.NumberTheory.Primes
a n = case factorise n of
    [(p, e)] -> unPrime p^(e-1) :: Int
    _ -> 0 -- Sebastian Karlsson, Nov 19 2021

Crossrefs

Cf. A003415, A003557, A055615, A063524, A069513, A100995, A120007, A129283, A246655.

Cf. also A000010, A000203, A069359, A300251, A319684, A327564, A349340, A349396, A349434, A349618, A349619, A349620, A349621.

Sequence in context: A343761 A336423 A128307 * A034369 A368096 A055277

Adjacent sequences: A349391 A349392 A349393 * A349395 A349396 A349397

Keyword

nonn

Author

Antti Karttunen, Nov 18 2021

Extensions

Added Sebastian Karlsson's formula as the new primary definition - Antti Karttunen, Nov 20 2021

Status

approved