A347961 -id:A347961 - cp's OEIS frontend

A347962 a(n) = A347961(A276086(n)), where A347961 is Dirichlet convolution of A342001 with itself.

0, 0, 0, 2, 1, 14, 0, 2, 2, 40, 20, 210, 1, 18, 20, 234, 120, 1040, 4, 52, 60, 576, 300, 2340, 10, 108, 128, 1090, 575, 4170, 0, 2, 2, 48, 24, 266, 2, 56, 60, 808, 410, 3760, 28, 378, 434, 4312, 2240, 17850, 92, 992, 1176, 10128, 5340, 39120, 208, 1954, 2370, 18592, 9920, 68410, 1, 22, 24, 330, 168, 1572, 28, 418

Offset: 0

Antti Karttunen, Sep 25 2021

Antti Karttunen, Table of n, a(n) for n = 0..11550
Index entries for sequences related to primorial base

Cf. A003415, A003557, A276086, A342001, A347961.

Cf. also the scatter plots of A342002, A347396 and A347126 (the most similar to this one).

PARI
```
A347962(n) = A347961(A276086(n));
```

a(n) = A347961(A276086(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).

Original entry on oeis.org

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

A349396 Dirichlet convolution of A342001 ({arithmetic derivative of n}/A003557(n)) with A055615 (Dirichlet inverse of n).

Original entry on oeis.org

0, 1, 1, 0, 1, 0, 1, -1, -1, 0, 1, -2, 1, 0, 0, -2, 1, -6, 1, -2, 0, 0, 1, -2, -3, 0, -3, -2, 1, 0, 1, -3, 0, 0, 0, 2, 1, 0, 0, -2, 1, 0, 1, -2, -6, 0, 1, -2, -5, -20, 0, -2, 1, -6, 0, -2, 0, 0, 1, 0, 1, 0, -6, -4, 0, 0, 1, -2, 0, 0, 1, 8, 1, 0, -20, -2, 0, 0, 1, -2, -5, 0, 1, 0, 0, 0, 0, -2, 1, 0, 0, -2, 0, 0, 0

Offset: 1

Antti Karttunen, Nov 18 2021

sign

Dirichlet convolution of this sequence with A000010 (Euler phi) is A346485.

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

Cf. A003415, A003557, A055615, A342001.
Cf. A346485, A347234,  A347235, A347395, A347954, A347959, A347961, A347963 for Dirichlet convolutions of A342001 with other sequences.
Cf. also A349394.

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));
A055615(n) = (n*moebius(n));
A349396(n) = sumdiv(n,d,A342001(n/d)*A055615(d));

a(n) = Sum_{d|n} A055615(d) * A342001(n/d).

A347963 Dirichlet convolution of A342001 with A003557.

Original entry on oeis.org

0, 1, 1, 3, 1, 7, 1, 7, 3, 9, 1, 18, 1, 11, 10, 15, 1, 18, 1, 24, 12, 15, 1, 40, 3, 17, 8, 30, 1, 54, 1, 31, 16, 21, 14, 45, 1, 23, 18, 54, 1, 68, 1, 42, 27, 27, 1, 84, 3, 24, 22, 48, 1, 48, 18, 68, 24, 33, 1, 132, 1, 35, 33, 63, 20, 96, 1, 60, 28, 90, 1, 99, 1, 41, 27, 66, 20, 110, 1, 114, 22, 45, 1, 168, 24, 47

Offset: 1

Antti Karttunen, Sep 24 2021

nonn

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

Cf. A003415, A003557, A342001.

Cf. also A347961.

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));
A347963(n) = sumdiv(n,d,A003557(n/d)*A342001(d));

a(n) = Sum_{d|n} A003557(d) * A342001(n/d).

cp's OEIS Frontend

A347962 a(n) = A347961(A276086(n)), where A347961 is Dirichlet convolution of A342001 with itself.

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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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A349396 Dirichlet convolution of A342001 ({arithmetic derivative of n}/A003557(n)) with A055615 (Dirichlet inverse of n).

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A347963 Dirichlet convolution of A342001 with A003557.

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