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 1-4 of 4 results.

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

Views

Author

Antti Karttunen, Nov 18 2021

Keywords

Comments

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

Links

Crossrefs

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.

Programs

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

Formula

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

Extensions

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

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)

A349619 Dirichlet convolution of A003415 with the Dirichlet inverse of A003557.

Original entry on oeis.org

0, 1, 1, 3, 1, 3, 1, 7, 5, 5, 1, 7, 1, 7, 6, 15, 1, 9, 1, 13, 8, 11, 1, 15, 9, 13, 19, 19, 1, 14, 1, 31, 12, 17, 10, 17, 1, 19, 14, 29, 1, 20, 1, 31, 24, 23, 1, 31, 13, 25, 18, 37, 1, 27, 14, 43, 20, 29, 1, 30, 1, 31, 34, 63, 16, 32, 1, 49, 24, 34, 1, 33, 1, 37, 34, 55, 16, 38, 1, 61, 65, 41, 1, 44, 20, 43, 30, 71
Offset: 1

Views

Author

Antti Karttunen, Nov 25 2021

Keywords

Links

Crossrefs

Cf. A003415, A003557, A349340.
Cf. also A349394, A349618, A349620, A349621.

Programs

  • Mathematica
    f1[p_, e_] := e/p; d[1] = 0; d[n_] := n*Plus @@ f1 @@@ FactorInteger[n]; f[p_, e_] := -(p - 1)^(e - 1); s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := DivisorSum[n, s[#]*d[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 25 2021 *)
  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A349340(n) = { my(f=factor(n)); prod(i=1, #f~, -((f[i,1]-1)^(f[i,2]-1))); };
A349619(n) = sumdiv(n,d,A003415(n/d)*A349340(d));

Formula

a(n) = Sum_{d|n} A003415(n/d) * A349340(d).

A349350 Dirichlet inverse of A057521, the powerful part of n.

Original entry on oeis.org

1, -1, -1, -3, -1, 1, -1, -1, -8, 1, -1, 3, -1, 1, 1, 5, -1, 8, -1, 3, 1, 1, -1, 1, -24, 1, -10, 3, -1, -1, -1, 7, 1, 1, 1, 24, -1, 1, 1, 1, -1, -1, -1, 3, 8, 1, -1, -5, -48, 24, 1, 3, -1, 10, 1, 1, 1, 1, -1, -3, -1, 1, 8, -3, 1, -1, -1, 3, 1, -1, -1, 8, -1, 1, 24, 3, 1, -1, -1, -5, 28, 1, -1, -3, 1, 1, 1, 1, -1, -8
Offset: 1

Views

Author

Antti Karttunen, Nov 18 2021

Keywords

Comments

Multiplicative because A057521 is.

Links

Crossrefs

Cf. A057521.
Cf. also A349340, A349442.

Programs

  • Mathematica
    f[p_, e_] := Module[{B = 1 + p - 2*p^2, C = Sqrt[1 + 2*p - 3*p^2]}, FullSimplify[((B - C)*(p - 1 + C)^(e - 1) - (B + C)*(p - 1 - C)^(e - 1))/(2^e*C)]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 24 2023 *)
  • PARI
    A057521(n) = { my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1)); }; \\ From A057521
memoA349350 = Map();
A349350(n) = if(1==n,1,my(v); if(mapisdefined(memoA349350,n,&v), v, v = -sumdiv(n,d,if(dA057521(n/d)*A349350(d),0)); mapput(memoA349350,n,v); (v)));

Formula

a(1) = 1; a(n) = -Sum_{d|n, d < n} A057521(n/d) * a(d).
Let p be a prime, B = 1 + p - 2*p^2 and C = sqrt(1 + 2*p - 3*p^2). Then the sequence is multiplicative with a(p^e) = ((B-C)*(p-1+C)^(e-1) - (B+C)*(p-1-C)^(e-1))/(2^e*C). - Sebastian Karlsson, Dec 02 2021
Showing 1-4 of 4 results.

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