cp's OEIS Frontend

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).

%I A342001 #41 Nov 04 2022 10:48:07
%S A342001 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 A342001 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 A342001 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 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).
%C A342001 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 A342001 Antti Karttunen, <a href="/A342001/b342001.txt">Table of n, a(n) for n = 1..16384</a>
%H A342001 Antti Karttunen, <a href="/A342001/a342001.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%F A342001 a(n) = A003415(n) / A003557(n).
%F A342001 For all n >= 0, a(A276086(n)) = A342002(n).
%F A342001 a(n) = A342414(n) * A342416(n) = A342459(n) * A342919(n). - _Antti Karttunen_, Apr 30 2022
%F A342001 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 A342001 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 A342001 Array[#1/#2 & @@ {If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &@ Abs[#], #/Times @@ FactorInteger[#][[All, 1]]} &, 91] (* _Michael De Vlieger_, Mar 11 2021 *)
%o A342001 (PARI)
%o A342001 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A342001 A003557(n) = (n/factorback(factorint(n)[, 1]));
%o A342001 A342001(n) = (A003415(n) / A003557(n));
%o A342001 (Python)
%o A342001 from math import prod
%o A342001 from sympy import factorint
%o A342001 def A342001(n):
%o A342001     q = prod(f:=factorint(n))
%o A342001     return sum(q*e//p for p, e in f.items()) # _Chai Wah Wu_, Nov 04 2022
%Y A342001 Cf. A003415, A003557, A341998, A342416, A342459.
%Y A342001 Cf. A342002 [= a(A276086(n))], A342463 [= a(A342456(n))], A351945 [= a(A181819(n))], A353571 [= a(A003961(n))].
%Y A342001 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 A342001 Cf. A007947.
%K A342001 nonn
%O A342001 1,4
%A A342001 _Antti Karttunen_, Feb 28 2021