cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

Antti Karttunen, Feb 28 2021

Keywords

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

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.

Programs

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

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

A349618 Dirichlet convolution of arithmetic derivative with A325126 [Dirichlet inverse of rad(n)].

Original entry on oeis.org

0, 1, 1, 2, 1, 0, 1, 6, 3, 0, 1, 2, 1, 0, 0, 14, 1, 6, 1, 2, 0, 0, 1, 2, 5, 0, 15, 2, 1, 0, 1, 34, 0, 0, 0, 18, 1, 0, 0, 2, 1, 0, 1, 2, 6, 0, 1, 6, 7, 20, 0, 2, 1, 6, 0, 2, 0, 0, 1, 0, 1, 0, 6, 78, 0, 0, 1, 2, 0, 0, 1, 42, 1, 0, 20, 2, 0, 0, 1, 6, 51, 0, 1, 0, 0, 0, 0, 2, 1, 0, 0, 2, 0, 0, 0, 10, 1, 42, 6, 50, 1, 0, 1, 2, 0
Offset: 1

Views

Author

Antti Karttunen, Nov 23 2021

Keywords

Links

Crossrefs

Cf. A003415, A007947, A325126.
Cf. also A349394, A349612.

Programs

  • Mathematica
    f[p_, e_] := e/p; d[1] = 0; d[n_] := n*Plus @@ f @@@ FactorInteger[n]; f2[p_, e_] := -p*(1 - p)^(e - 1); s[1] = 1; s[n_] := Times @@ f2 @@@ FactorInteger[n]; a[n_] := DivisorSum[n, d[#]*s[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 23 2021 *)
  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
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)));
A349618(n) = sumdiv(n,d,A003415(d)*A325126(n/d));

Formula

a(n) = Sum_{d|n} A003415(d) * A325126(n/d).

A325126 a(1) = 1; a(n) = -Sum_{d|n, dA007947.

Original entry on oeis.org

1, -2, -3, 2, -5, 6, -7, -2, 6, 10, -11, -6, -13, 14, 15, 2, -17, -12, -19, -10, 21, 22, -23, 6, 20, 26, -12, -14, -29, -30, -31, -2, 33, 34, 35, 12, -37, 38, 39, 10, -41, -42, -43, -22, -30, 46, -47, -6, 42, -40, 51, -26, -53, 24, 55, 14, 57, 58, -59, 30
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 04 2019

Keywords

Comments

Dirichlet inverse of A007947.
Moebius transform of A125131.

Links

Crossrefs

Cf. A007947, A008836, A125131, A326297.
Cf. also A327564, A332732, A349340, A349612, A349618.

Programs

  • Mathematica
    a[n_] := If[n == 1, n, -Sum[If[d < n, Last[Select[Divisors[n/d], SquareFreeQ]] a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 1, 60}]
f[p_, e_] := -p*(1 - p)^(e - 1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 14 2020 *)
  • PARI
    rad(n) = factorback(factorint(n)[, 1]); \\ A007947
lista(nn) = {my(va=vector(nn)); va[1] = 1; for (n=2, nn, va[n] = -sumdiv(n, d, if (dMichel Marcus, Jun 01 2020

Formula

G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} rad(k) * A(x^k).
From Isaac Saffold, May 30 2020: (Start)
a(n) = A008836(n)*A326297(n)*A007947(n).
Proof:
Define lambda(n) := A008836(n); h(n) := A326297(n); rad(n) := A007947(n).
As lambda(n), h(n), and rad(n) are multiplicative, the identity needs only to be proved for prime power n.
It is clear that the identity holds for n = 1 = p^0. For a given nonnegative integer k, assume the identity holds for all v such that 0 <= v <= k. Then, by the recursive formula for Dirichlet inverses,
a(p^(k+1)) = -Sum_{v=0..k} lambda(p^v)*h(p^v)*rad(p^v)*rad(p^(k+1-v))
= -p * (1 + p*Sum_{v=1..k}((-1)^v * (p-1)^(v-1)))
= -p * (1 - p*Sum_{v=0..(k-1)}((1 - p)^v))
= -p * (1 - p*(((1-p)^k - 1) / -p))
= -p * (1-p)^k
= (-1)^(k+1) * (p-1)^k * p
= lambda(p^(k+1)) * h(p^(k+1)) * rad(p^(k+1))
Thus the identity holds for p^(k+1), k >= 0.
As k is arbitrary and the identity holds for p^0, it holds for the prime powers, and thus for all positive integers. Q.E.D. (End)
Showing 1-3 of 3 results.

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