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.

A319131 a(n) = Sum_{d|n} Sum_{p|d, p prime} p.

Original entry on oeis.org

0, 2, 3, 4, 5, 10, 7, 6, 6, 14, 11, 17, 13, 18, 16, 8, 17, 18, 19, 23, 20, 26, 23, 24, 10, 30, 9, 29, 29, 40, 31, 10, 28, 38, 24, 30, 37, 42, 32, 32, 41, 48, 43, 41, 27, 50, 47, 31, 14, 26, 40, 47, 53, 26, 32, 40, 44, 62, 59, 64, 61, 66, 33, 12, 36, 64, 67, 59, 52, 56
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 11 2018

Keywords

Comments

Inverse Möbius transform of A008472.

Examples

			a(12) = 13 as 12 has 6 divisors and 2 * 6 * (2/3) + 3 * 6 * (1/2) = 17. - _David A. Corneth_, Oct 08 2019

Links

Crossrefs

Cf. A000005, A007429, A008472, A061397, A062799, A319132.

Programs

  • Magma
    [0] cat  [&+[&+[PrimeDivisors(d)[i]:i in [1..#PrimeDivisors(d)]]:d in Set(Divisors(n)) diff {1}]:n in [2..70]]; // Marius A. Burtea, Oct 08 2019
  • Magma
    [0] cat [&+[p*#Divisors(n div p):p in PrimeDivisors(n)]:n in [2..70]]; // Marius A. Burtea, Oct 08 2019 (According to the formula given by Ridouane Oudra)
  • Maple
    with(numtheory): seq(add(p*tau(n/p), p in factorset(n)), n=1..80); # Ridouane Oudra, Oct 08 2019
  • Mathematica
    Table[Sum[Total[Select[Divisors[d], PrimeQ]], {d, Divisors[n]}], {n, 70}]
nmax = 70; Rest[CoefficientList[Series[Sum[DivisorSum[k, # &, PrimeQ[#] &] x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 70; Rest[CoefficientList[Series[-Log[Product[(1 - x^k)^(DivisorSum[k, # &, PrimeQ[#] &]/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
  • PARI
    a(n) = sumdiv(n, d, my(f=factor(d)); vecsum(f[,1])); \\ Michel Marcus, Oct 08 2019
  • PARI
    a(n) = my(f = factor(n), nd = numdiv(f)); sum(i = 1, #f~, f[i, 1] * nd / (f[i, 2] + 1) * f[i, 2]) \\ David A. Corneth, Oct 08 2019

Formula

G.f.: Sum_{k>=1} A008472(k)*x^k/(1 - x^k).
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(A008472(k)/k)) = Sum_{n>=1} a(n)*x^n/n.
a(p^k) = p*k, where p is a prime.
a(n) = Sum_{p|n} p*tau(n/p), where p is a prime and tau(n) = A000005(n). - Ridouane Oudra, Oct 08 2019
a(n) = Sum_{p|n} p*tau(n)*(e_p-1)/(e_p) where e_p is the exponent of p in the factorization of n. - David A. Corneth, Oct 08 2019
a(n) = Sum_{d|n} sopf(d). - Wesley Ivan Hurt, May 23 2021

A328485 Dirichlet g.f.: zeta(s)^2 * zeta(s-1) / zeta(2*s-1).

Original entry on oeis.org

1, 4, 5, 9, 7, 20, 9, 18, 15, 28, 13, 45, 15, 36, 35, 35, 19, 60, 21, 63, 45, 52, 25, 90, 33, 60, 43, 81, 31, 140, 33, 68, 65, 76, 63, 135, 39, 84, 75, 126, 43, 180, 45, 117, 105, 100, 49, 175, 59, 132, 95, 135, 55, 172, 91, 162, 105, 124, 61, 315, 63, 132, 135, 133, 105
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 16 2019

Keywords

Comments

Inverse Moebius transform of A034448.
Dirichlet convolution of A055615 with A064840.

Links

Crossrefs

Cf. A000005, A000203, A007429, A008683, A034448, A048691, A055615, A064840, A319131, A319132.

Programs

  • Maple
    with(numtheory):
a:= n-> add(mobius(d)*tau(n/d)*sigma(n/d)*d, d=divisors(n)):
seq(a(n), n=1..70);  # Alois P. Heinz, Oct 16 2019
  • Mathematica
    Table[n DivisorSum[n, MoebiusMu[n/#] DivisorSigma[0, #] DivisorSigma[1, #]/# &], {n, 1, 65}]
nmax = 65; CoefficientList[Series[Sum[DivisorSum[k, # &, CoprimeQ[#, k/#] &] x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
f[p_, e_] := (p^(e + 1) - p)/(p - 1) + e + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 10 2023 *)
  • PARI
    a(n) = {my(f = factor(n), p = f[, 1], e = f[, 2]); prod(i = 1, #p, (p[i]^(e[i] + 1) - p[i])/(p[i] - 1) + e[i] + 1);} \\ Amiram Eldar, Feb 10 2023

Formula

G.f.: Sum_{k>=1} usigma(k) * x^k / (1 - x^k), where usigma = A034448.
a(n) = Sum_{d|n} usigma(d).
a(n) = n * Sum_{d|n} mu(n/d) * tau(d) * sigma(d) / d, where mu = A008683, tau = A000005 and sigma = A000203.
Sum_{k=1..n} a(k) ~ Pi^4 * n^2 / (72 * zeta(3)). - Vaclav Kotesovec, Oct 17 2019
From Amiram Eldar, Feb 10 2023: (Start)
a(n) = Sum_{d|n} Sum_{d'|n, gcd(d, d')=1} d'.
Multiplicative with a(p^e) = (p^(e+1)-p)/(p-1) + e + 1. (End)

A343442 If n = Product (p_j^k_j) then a(n) = Product (p_j + 2), with a(1) = 1.

Original entry on oeis.org

1, 4, 5, 4, 7, 20, 9, 4, 5, 28, 13, 20, 15, 36, 35, 4, 19, 20, 21, 28, 45, 52, 25, 20, 7, 60, 5, 36, 31, 140, 33, 4, 65, 76, 63, 20, 39, 84, 75, 28, 43, 180, 45, 52, 35, 100, 49, 20, 9, 28, 95, 60, 55, 20, 91, 36, 105, 124, 61, 140, 63, 132, 45, 4, 105, 260, 69, 76, 125, 252
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 15 2021

Keywords

Links

Crossrefs

Cf. A000203, A001615, A007429, A007947, A008683, A048250, A062953, A063441, A074816, A092261, A107758, A107759, A319132, A335005.

Programs

  • Mathematica
    a[1] = 1; a[n_] := Times @@ ((#[[1]] + 2) & /@ FactorInteger[n]); Table[a[n], {n, 70}]
nmax = 70; CoefficientList[Series[Sum[MoebiusMu[k]^2 DivisorSigma[1, k] x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
  • PARI
    a(n) = sumdiv(n, d, moebius(d)^2 * sigma(d)) \\ Andrew Howroyd, Apr 15 2021

Formula

G.f.: Sum_{k>=1} mu(k)^2 * sigma(k) * x^k / (1 - x^k), where mu = A008683 and sigma = A000203.
a(n) = Sum_{d|n} mu(d)^2 * sigma(d).
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/(12*zeta(3)) = 0.684216... (A335005). - Amiram Eldar, Nov 13 2022
a(n) = Sum_{d|n} mu(d)^2*psi(d), where psi is A001615. - Ridouane Oudra, Jul 24 2025
Showing 1-3 of 3 results.

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