A342001 Arithmetic derivative without its inherited divisor; the arithmetic derivative of n divided by A003557(n), which is a common divisor of both n and A003415(n).

0, 1, 1, 2, 1, 5, 1, 3, 2, 7, 1, 8, 1, 9, 8, 4, 1, 7, 1, 12, 10, 13, 1, 11, 2, 15, 3, 16, 1, 31, 1, 5, 14, 19, 12, 10, 1, 21, 16, 17, 1, 41, 1, 24, 13, 25, 1, 14, 2, 9, 20, 28, 1, 9, 16, 23, 22, 31, 1, 46, 1, 33, 17, 6, 18, 61, 1, 36, 26, 59, 1, 13, 1, 39, 11, 40, 18, 71, 1, 22, 4, 43, 1, 62, 22, 45, 32, 35, 1, 41, 20

Offset

1,4

Comments

See also the scatter plot of A342002 that seems to reveal some interesting internal structure in this sequence, not fully explained by the regularity of primorial base expansion used in the latter sequence. - Antti Karttunen, May 09 2022

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Formula

a(n) = A003415(n) / A003557(n).

For all n >= 0, a(A276086(n)) = A342002(n).

a(n) = A342414(n) * A342416(n) = A342459(n) * A342919(n). - Antti Karttunen, Apr 30 2022

Dirichlet g.f.: Dirichlet g.f. of A007947 * Sum_{p prime} p^s/((p^s-1)*(p^s+p-1)) = zeta(s) * Product_{p prime} (1+p^(1-s)-p^(-s)) * Sum_{p prime} p^s/((p^s-1)*(p^s+p-1)). - Sebastian Karlsson, May 05 2022

Sum_{k=1..n} a(k) ~ c * A065464 * Pi^2 * n^2 / 12, where c = Sum_{j>=2} (1/2 + (-1)^j * (Fibonacci(j) - 1/2))*PrimeZetaP(j) = 0.4526952873143153104685540856936425315834753528741817723313791528384... - Vaclav Kotesovec, May 09 2022

Mathematica

Array[#1/#2 & @@ {If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &@ Abs[#], #/Times @@ FactorInteger[#][[All, 1]]} &, 91] (* Michael De Vlieger, Mar 11 2021 *)

Prog

(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A342001(n) = (A003415(n) / A003557(n));
(Python)
from math import prod
from sympy import factorint
def A342001(n):
    q = prod(f:=factorint(n))
    return sum(q*e//p for p, e in f.items()) # Chai Wah Wu, Nov 04 2022

Crossrefs

Cf. A003415, A003557, A341998, A342416, A342459.

Cf. A342002 [= a(A276086(n))], A342463 [= a(A342456(n))], A351945 [= a(A181819(n))], A353571 [= a(A003961(n))].

Cf. A346485 (Möbius transform), A347395 (convolution with Liouville's lambda), A347961 (with itself), and A347234, A347235, A347954, A347959, A347963, A349396, A349612 (for convolutions with other sequences).

Cf. A007947.

Sequence in context: A079168 A332085 A055205 * A347957 A377801 A369038

Adjacent sequences: A341998 A341999 A342000 * A342002 A342003 A342004

Keyword

nonn

Author

Antti Karttunen, Feb 28 2021

Status

approved