%I #41 Nov 04 2022 10:48:07

%S 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,

%T 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,

%U 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

%N 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).

%C 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

%H Antti Karttunen, <a href="/A342001/b342001.txt">Table of n, a(n) for n = 1..16384</a>

%H Antti Karttunen, <a href="/A342001/a342001.txt">Data supplement: n, a(n) computed for n = 1..65537</a>

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

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

%F a(n) = A342414(n) * A342416(n) = A342459(n) * A342919(n). - _Antti Karttunen_, Apr 30 2022

%F 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

%F 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

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

%o (PARI)

%o A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

%o A003557(n) = (n/factorback(factorint(n)[, 1]));

%o A342001(n) = (A003415(n) / A003557(n));

%o (Python)

%o from math import prod

%o from sympy import factorint

%o def A342001(n):

%o q = prod(f:=factorint(n))

%o return sum(q*e//p for p, e in f.items()) # _Chai Wah Wu_, Nov 04 2022

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

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

%Y 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).

%Y Cf. A007947.

%K nonn

%O 1,4

%A _Antti Karttunen_, Feb 28 2021