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 1-5 of 5 results.

A349434 Dirichlet convolution of A129283 (n + its arithmetic derivative) with A349337 (Dirichlet inverse of A230593).

Original entry on oeis.org

1, 0, 0, 2, 0, 0, 0, 2, 3, 0, 0, -2, 0, 0, 0, 6, 0, -3, 0, -2, 0, 0, 0, 0, 5, 0, 6, -2, 0, 0, 0, 10, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, -2, -3, 0, 0, -6, 7, -5, 0, -2, 0, -3, 0, 0, 0, 0, 0, 4, 0, 0, -3, 22, 0, 0, 0, -2, 0, 0, 0, -5, 0, 0, -5, -2, 0, 0, 0, -6, 21, 0, 0, 4, 0, 0, 0, 0, 0, 6, 0, -2, 0, 0, 0, -4, 0, -7, -3, 7
Offset: 1

Views

Author

Antti Karttunen, Nov 17 2021

Keywords

Comments

Dirichlet convolution of this sequence with A349338 is A348976.

Links

Crossrefs

Cf. A003415, A129283, A230593, A349337, A349435 (Dirichlet inverse), A349436 (sum with it).
Cf. also A348976, A349338.

Programs

  • Mathematica
    s[n_] := n * DivisorSum[n, 1/# &, !CompositeQ[#] &]; sinv[1] = 1; sinv[n_] := sinv[n] = -DivisorSum[n, sinv[#] * s[n/#] &, # < n &]; f[p_, e_] := e/p; d[1] = 1; d[n_] := n*(1 + Plus @@ f @@@ FactorInteger[n]); a[n_] := DivisorSum[n, sinv[#] * d[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 18 2021 *)
  • PARI
    up_to = 20000;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A129283(n) = (n+A003415(n));
A230593(n) = sumdiv(n, d, ((1==d)||isprime(d))*(n/d));
v349337 = DirInverseCorrect(vector(up_to,n,A230593(n)));
A349337(n) = v349337[n];
A349434(n) = sumdiv(n,d,A129283(n/d)*A349337(d));

Formula

a(n) = Sum_{d|n} A129283(n/d) * A349337(d).

A349435 Dirichlet convolution of A230593 with A347084, which is Dirichlet inverse of {n + its arithmetic derivative}.

Original entry on oeis.org

1, 0, 0, -2, 0, 0, 0, -2, -3, 0, 0, 2, 0, 0, 0, -2, 0, 3, 0, 2, 0, 0, 0, 0, -5, 0, -6, 2, 0, 0, 0, -2, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 2, 3, 0, 0, -2, -7, 5, 0, 2, 0, 3, 0, 0, 0, 0, 0, -4, 0, 0, 3, -2, 0, 0, 0, 2, 0, 0, 0, 5, 0, 0, 5, 2, 0, 0, 0, -2, -12, 0, 0, -4, 0, 0, 0, 0, 0, -6, 0, 2, 0, 0, 0, -4, 0, 7, 3
Offset: 1

Views

Author

Antti Karttunen, Nov 17 2021

Keywords

Comments

Dirichlet convolution of this sequence with A348976 is A349338.
The positions of records start as: 1, 12, 18, 36, 100, 108, 196, 225, 324, 441, 500, 1125, 1372, 2500, 5000, 5324, 8575, 8788, 9604, 12500, 19652, etc.

Links

Crossrefs

Cf. A003415, A129283, A230593, A347084, A349434 (Dirichlet inverse), A349436 (sum with it).
Cf. also A348976, A349338.

Programs

  • Mathematica
    s[n_] := n * DivisorSum[n, 1/# &, !CompositeQ[#] &]; f[p_, e_] := e/p; d[1] = 1; d[n_] := n*(1 + Plus @@ f @@@ FactorInteger[n]); dinv[1] = 1; dinv[n_] := dinv[n] = -DivisorSum[n, dinv[#] * d[n/#] &, # < n &]; a[n_] := DivisorSum[n, s[#] * dinv[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 18 2021 *)
  • PARI
    up_to = 20000;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A230593(n) = sumdiv(n, d, ((1==d)||isprime(d))*(n/d));
v347084 = DirInverseCorrect(vector(up_to,n,n+A003415(n)));
A347084(n) = v347084[n];
A349435(n) = sumdiv(n,d,A230593(n/d)*A347084(d));

Formula

a(n) = Sum_{d|n} A230593(n/d) * A347084(d).

A349337 Dirichlet inverse of A230593.

Original entry on oeis.org

1, -3, -4, 3, -6, 13, -8, -3, 4, 19, -12, -16, -14, 25, 25, 3, -18, -17, -20, -22, 33, 37, -24, 19, 6, 43, -4, -28, -30, -87, -32, -3, 49, 55, 49, 33, -38, 61, 57, 25, -42, -113, -44, -40, -29, 73, -48, -22, 8, -25, 73, -46, -54, 21, 73, 31, 81, 91, -60, 125, -62, 97, -37, 3, 85, -165, -68, -58, 97, -163, -72, -52
Offset: 1

Views

Author

Antti Karttunen, Nov 15 2021

Keywords

Comments

Coincides with A347084 on all squarefree numbers (A005117), but also on n=81, where a(81) = A347084(81) = 4. Question: Are there any other such numbers?

Links

Crossrefs

Cf. A005117, A069359, A230593.
Cf. also A347084, A349434, A349435.

Programs

  • PARI
    up_to = 20000;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA230593(n) = sumdiv(n, d, ((1==d)||isprime(d))*(n/d));
v349337 = DirInverseCorrect(vector(up_to,n,A230593(n)));
A349337(n) = v349337[n];

Formula

For n > 1, a(n) = -Sum_{d|n, 1A230593(d) * A349337(n/d).

A069359 a(n) = n * Sum_{p|n} 1/p where p are primes dividing n.

Original entry on oeis.org

0, 1, 1, 2, 1, 5, 1, 4, 3, 7, 1, 10, 1, 9, 8, 8, 1, 15, 1, 14, 10, 13, 1, 20, 5, 15, 9, 18, 1, 31, 1, 16, 14, 19, 12, 30, 1, 21, 16, 28, 1, 41, 1, 26, 24, 25, 1, 40, 7, 35, 20, 30, 1, 45, 16, 36, 22, 31, 1, 62, 1, 33, 30, 32, 18, 61, 1, 38, 26, 59, 1, 60, 1, 39, 40, 42, 18, 71, 1, 56
Offset: 1

Views

Author

Benoit Cloitre, Apr 15 2002

Keywords

Comments

Coincides with arithmetic derivative on squarefree numbers: a(A005117(n)) = A068328(n) = A003415(A005117(n)). - Reinhard Zumkeller, Jul 20 2003, Clarified by Antti Karttunen, Nov 15 2019
a(n) = n-1 iff n = 1 or n is a primary pseudoperfect number A054377. - Jonathan Sondow, Apr 16 2014
a(1) = 0 by the standard convention for empty sums.
“Seva” on the MathOverflow link asks if the iterates of this sequence are all eventually 0. - Charles R Greathouse IV, Feb 15 2019

Examples

			a(12) = 10 because the prime divisors of 12 are 2 and 3 so we have: 12/2 + 12/3 = 6 + 4 = 10. - _Geoffrey Critzer_, Mar 17 2015

Links

Crossrefs

Cf. A003415, A005117, A068328, A010051, A000027, A054377, A180253, A230593, A292786, A306369, A326690, A329029, A329350, A329352.
Cf. A322068 (partial sums), A323599 (Inverse Möbius transform).
Sequences of the form n^k * Sum_{p|n, p prime} 1/p^k for k = 0..10: A001221 (k=0), this sequence (k=1), A322078 (k=2), A351242 (k=3), A351244 (k=4), A351245 (k=5), A351246 (k=6), A351247 (k=7), A351248 (k=8), A351249 (k=9), A351262 (k=10).

Programs

  • Magma
    [0] cat [n*&+[1/p: p in PrimeDivisors(n)]:n in [2..80]]; // Marius A. Burtea, Jan 21 2020
  • Maple
    A069359 := n -> add(n/d, d = select(isprime, numtheory[divisors](n))):
seq(A069359(i), i = 1..20); # Peter Luschny, Jan 31 2012
# second Maple program:
a:= n-> n*add(1/i[1], i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 23 2019
  • Mathematica
    f[list_, i_] := list[[i]]; nn = 100; a = Table[n, {n, 1, nn}]; b =
Table[If[PrimeQ[n], 1, 0], {n, 1, nn}]; Table[DirichletConvolve[f[a, n], f[b, n], n, m], {m, 1, nn}] (* Geoffrey Critzer, Mar 17 2015 *)
  • PARI
    a(n) = n*sumdiv(n, d, isprime(d)/d); \\ Michel Marcus, Mar 18 2015
  • PARI
    a(n) = my(ps=factor(n)[,1]~);sum(k=1,#ps,n\ps[k]) \\ Franklin T. Adams-Watters, Apr 09 2015
  • Python
    from sympy import primefactors
def A069359(n): return sum(n//p for p in primefactors(n)) # Chai Wah Wu, Feb 05 2022
  • Sage
    def A069359(n) :
    D = filter(is_prime, divisors(n))
    return add(n/d for d in D)
print([A069359(i) for i in (1..20)]) # Peter Luschny, Jan 31 2012

Formula

G.f.: Sum(x^p(j)/(1-x^p(j))^2,j>=1), where p(j) is the j-th prime. - Vladeta Jovovic, Mar 29 2006
a(n) = A230593(n) - n. a(n) = A010051(n) (*) A000027(n), where operation (*) denotes Dirichlet convolution, that is, convolution of type: a(n) = Sum_(d|n) b(d) * c(n/d) = Sum_{d|n} A010051(d) * A000027(n/d). - Jaroslav Krizek, Nov 07 2013
a(A054377(n)) = A054377(n) - 1. - Jonathan Sondow, Apr 16 2014
Dirichlet g.f.: zeta(s - 1)*primezeta(s). - Geoffrey Critzer, Mar 17 2015
Sum_{k=1..n} a(k) ~ A085548 * n^2 / 2. - Vaclav Kotesovec, Feb 04 2019
From Antti Karttunen, Nov 15 2019: (Start)
a(n) = Sum_{d|n} A008683(n/d)*A323599(d).
a(n) = A003415(n) - A329039(n) = A230593(n) - n = A306369(n) - A000010(n).
a(n) = A276085(A329350(n)) = A048675(A329352(n)).
a(A276086(n)) = A329029(n), a(A328571(n)) = A329031(n).
(End)
a(n) = Sum_{d|n} A000010(d) * A001221(n/d). - Torlach Rush, Jan 21 2020
a(n) = Sum_{k=1..n} omega(gcd(n, k)). - Ilya Gutkovskiy, Feb 21 2020
a(p^k) = p^(k-1) for p prime and k>=1. - Wesley Ivan Hurt, Jul 15 2025

A349338 Dirichlet convolution of A000010 (Euler totient phi) with A080339 (characteristic function of noncomposite numbers).

Original entry on oeis.org

1, 2, 3, 3, 5, 5, 7, 6, 8, 9, 11, 8, 13, 13, 14, 12, 17, 14, 19, 14, 20, 21, 23, 16, 24, 25, 24, 20, 29, 22, 31, 24, 32, 33, 34, 22, 37, 37, 38, 28, 41, 32, 43, 32, 38, 45, 47, 32, 48, 44, 50, 38, 53, 42, 54, 40, 56, 57, 59, 36, 61, 61, 54, 48, 64, 52, 67, 50, 68, 58, 71, 44, 73, 73, 68, 56, 76, 62, 79, 56, 72, 81
Offset: 1

Views

Author

Antti Karttunen, Nov 17 2021

Keywords

Comments

Möbius transform of A230593.
The number of integers k from 1 to n such that gcd(n, k) is a noncomposite number. - Amiram Eldar, Jun 21 2025

Links

Crossrefs

Cf. A000010, A005117, A008683, A069359, A080339, A085548, A117494, A230593, A348976, A349435.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, Boole[!CompositeQ[#]] * EulerPhi[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 17 2021 *)
  • PARI
    A349338(n) = sumdiv(n, d, eulerphi(n/d)*((1==d)||isprime(d)));
  • PARI
    a(n) = {my(f = factor(n), p = f[,1], e = f[,2]); n * vecprod(apply(x -> 1-1/x, p)) * (1 + vecsum(apply(x -> 1/x, p - vector(#e, i, e[i] == 1)~)));} \\ Amiram Eldar, Jun 21 2025

Formula

a(n) = Sum_{d|n} A000010(n/d) * A080339(d).
a(n) = Sum_{d|n} A008683(n/d) * A230593(d).
a(n) = Sum_{d|n} A349435(n/d) * A348976(d).
a(n) = A000010(n) + A117494(n). [Because A117494 is the Möbius transform of A069359]
For all n >= 1, a(A005117(n)) = A348976(A005117(n)).
Sum_{k=1..n} a(k) ~ 3 * (1 + A085548) * n^2 / Pi^2. - Vaclav Kotesovec, Nov 20 2021
Showing 1-5 of 5 results.

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