A349396 -id:A349396 - 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).

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

Antti Karttunen, Feb 28 2021

nonn

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

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

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.

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

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));

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

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

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).

Original entry on oeis.org

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

Antti Karttunen, Nov 18 2021

nonn

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

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.

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.

import Math.NumberTheory.Primes
a n = case factorise n of
    [(p,e)] -> unPrime p^(e-1) :: Int
     -> 0 -- _Sebastian Karlsson, Nov 19 2021

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 *)

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));

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

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

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

A349612 Dirichlet convolution of A342001 [{arithmetic derivative of n}/A003557(n)] with A325126 [Dirichlet inverse of rad(n)].

Original entry on oeis.org

0, 1, 1, 0, 1, 0, 1, 1, -1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, -3, 0, 3, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, -5, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, -5, 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

Antti Karttunen, Nov 23 2021

sign

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

Cf. A003415, A003557, A007947, A342001, A325126.

Cf. also A349394, A349396, A349618.

f[p_, e_] := e/p; d[1] = 0; d[n_] := n * Plus @@ f @@@ FactorInteger[n]; f1[p_, e_] := p^(e-1); s1[1] = 1; s1[n_] := Times @@ f1 @@@ FactorInteger[n]; f2[p_, e_] := -p*(1 - p)^(e - 1); s2[1] = 1; s2[n_] := Times @@ f2 @@@ FactorInteger[n]; a[n_] := DivisorSum[n, d[#]*s2[n/#]/s1[#] &]; Array[a, 100] (* Amiram Eldar, Nov 23 2021 *)

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));
A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947
memoA325126 = Map();
A325126(n) = if(1==n,1,my(v); if(mapisdefined(memoA325126,n,&v), v, v = -sumdiv(n,d,if(dA007947(n/d)*A325126(d),0)); mapput(memoA325126,n,v); (v)));
A349612(n) = sumdiv(n,d,A342001(d)*A325126(n/d));

a(n) = Sum_{d|n} A342001(d) * A325126(n/d).
If p prime, a(p) = 1. - Bernard Schott, Nov 28 2021
Dirichlet g.f.: Sum_{p prime} p^s/((p^s-1)*(p^s+p-1)). - Sebastian Karlsson, May 05 2022

A369455 Dirichlet convolution of A083345 with A055615 (Dirichlet inverse of n), where A083345(n) = (n'/gcd(n,n')) and n' is the arithmetic derivative of n.

Original entry on oeis.org

0, 1, 1, -1, 1, 0, 1, 1, -1, 0, 1, -3, 1, 0, 0, -4, 1, -6, 1, -3, 0, 0, 1, 0, -3, 0, -5, -3, 1, 0, 1, 1, 0, 0, 0, 9, 1, 0, 0, 0, 1, 0, 1, -3, -6, 0, 1, -3, -5, -20, 0, -3, 1, -8, 0, 0, 0, 0, 1, 0, 1, 0, -6, -7, 0, 0, 1, -3, 0, 0, 1, -6, 1, 0, -20, -3, 0, 0, 1, -3, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, -3, 0, 0, 0, 0, 1, -42, -6, 29

Offset: 1

Antti Karttunen, Jan 27 2024

sign

Dirichlet convolution of this sequence with A000010 (Euler phi) is A369068.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A055615, A083345.

Cf. also A349394, A349396, A369068, A369069.

A055615(n) = (n*moebius(n));
A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
A369455(n) = sumdiv(n, d, A083345(n/d)*A055615(d));

a(n) = Sum_{d|n} A055615(n/d)*A083345(d).

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).

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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).

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A349612 Dirichlet convolution of A342001 [{arithmetic derivative of n}/A003557(n)] with A325126 [Dirichlet inverse of rad(n)].

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A369455 Dirichlet convolution of A083345 with A055615 (Dirichlet inverse of n), where A083345(n) = (n'/gcd(n,n')) and n' is the arithmetic derivative of n.

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