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 1-6 of 6 results.

A319132 a(n) = Sum_{d|n} Sum_{j|d} mu(j)^2*j, where mu = Möbius function (A008683).

Original entry on oeis.org

1, 4, 5, 7, 7, 20, 9, 10, 9, 28, 13, 35, 15, 36, 35, 13, 19, 36, 21, 49, 45, 52, 25, 50, 13, 60, 13, 63, 31, 140, 33, 16, 65, 76, 63, 63, 39, 84, 75, 70, 43, 180, 45, 91, 63, 100, 49, 65, 17, 52, 95, 105, 55, 52, 91, 90, 105, 124, 61, 245, 63, 132, 81, 19, 105, 260, 69, 133, 125, 252
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 11 2018

Keywords

Comments

Inverse Möbius transform of A048250.

Links

Crossrefs

Cf. A007429, A008683, A048250, A048691, A055615, A072691, A319131.

Programs

  • Magma
    [&+[&+[MoebiusMu(j)^2*j:j in Divisors(d)]:d in Divisors(n)]:n in [1..70]]; // Marius A. Burtea, Nov 13 2019
  • Magma
    [&+[MoebiusMu(d)^2*d*NumberOfDivisors(n div d):d in Divisors(n)]:n in [1..70]]; // Marius A. Burtea, Nov 13 2019
  • Maple
    with(numtheory): seq(add(mobius(d)^2*d*tau(n/d), d in divisors(n)), n=1..70); # Ridouane Oudra, Nov 13 2019
  • Mathematica
    Table[Sum[Sum[MoebiusMu[j]^2 j, {j, Divisors[d]}], {d, Divisors[n]}], {n, 70}]
nmax = 70; Rest[CoefficientList[Series[Sum[DivisorSum[k, # &, SquareFreeQ[#] &] x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 70; Rest[CoefficientList[Series[-Log[Product[(1 - x^k)^(DivisorSum[k, # &, SquareFreeQ[#] &]/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
f[p_, e_] := (p + 1)*e + 1; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 25 2020 *)
  • PARI
    a(n) = sumdiv(n, d, moebius(d)^2*d*numdiv(n/d)); \\ Michel Marcus, Nov 13 2019; corrected Jun 13 2022

Formula

G.f.: Sum_{k>=1} A048250(k)*x^k/(1 - x^k).
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(A048250(k)/k)) = Sum_{n>=1} a(n)*x^n/n.
a(p^k) = (p + 1)*k + 1, where p is a prime.
a(n) = Sum_{d|n} mu(d)^2*d*tau(n/d). - Ridouane Oudra, Nov 13 2019
Multiplicative with a(p^e) = (p+1)*e+1. - Amiram Eldar, Oct 25 2020
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/12 = 0.822467... (A072691). - Amiram Eldar, Nov 13 2022
Dirichlet g.f.: zeta(s)^2*zeta(s-1)/zeta(2*s-2). - Amiram Eldar, Jan 03 2023

A369894 a(n) = n * Sum_{p|n, p prime} tau(n/p) / p.

Original entry on oeis.org

0, 1, 1, 4, 1, 10, 1, 12, 6, 14, 1, 36, 1, 18, 16, 32, 1, 51, 1, 52, 20, 26, 1, 104, 10, 30, 27, 68, 1, 124, 1, 80, 28, 38, 24, 180, 1, 42, 32, 152, 1, 164, 1, 100, 87, 50, 1, 272, 14, 115, 40, 116, 1, 216, 32, 200, 44, 62, 1, 432, 1, 66, 111, 192, 36, 244, 1, 148
Offset: 1

Views

Author

Wesley Ivan Hurt, Feb 04 2024

Keywords

Links

Crossrefs

Cf. A000005 (tau), A062799, A319131, A369893.

Programs

  • Mathematica
    Table[n*DivisorSum[n, DivisorSigma[0, n/#]/# &, PrimeQ[#] &], {n, 80}]
  • PARI
    A369894(n) = if(1==n, 0, my(f=factor(n)); n*sum(i=1, #f~, (numdiv(n/f[i, 1])/f[i,1]))); \\ Antti Karttunen, Jan 22 2025

Formula

a(p^k) = k*p^(k-1), for prime p and k >= 1. - Wesley Ivan Hurt, Jun 26 2024

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)

A349712 a(n) = Sum_{d|n} sopf(d) * sopf(n/d).

Original entry on oeis.org

0, 0, 0, 4, 0, 12, 0, 8, 9, 20, 0, 32, 0, 28, 30, 12, 0, 42, 0, 48, 42, 44, 0, 52, 25, 52, 18, 64, 0, 124, 0, 16, 66, 68, 70, 87, 0, 76, 78, 76, 0, 164, 0, 96, 78, 92, 0, 72, 49, 90, 102, 112, 0, 72, 110, 100, 114, 116, 0, 234, 0, 124, 102, 20, 130, 244, 0, 144, 138, 236, 0, 132, 0, 148, 110
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 26 2021

Keywords

Comments

Dirichlet convolution of A008472 with itself.

Links

Crossrefs

Cf. A008472, A034761, A061397, A070288, A319131, A349711.

Programs

  • Mathematica
    sopf[n_] := DivisorSum[n, # &, PrimeQ[#] &]; a[n_] := Sum[sopf[d] sopf[n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 75}]
  • PARI
    sopf(n) = vecsum(factor(n)[, 1]); \\ A008472
a(n) = sumdiv(n, d, sopf(d)*sopf(n/d)); \\ Michel Marcus, Nov 26 2021
  • Python
    from sympy import divisors, factorint
def sopf(n): return sum(factorint(n))
def a(n): return sum(sopf(d)*sopf(n//d) for d in divisors(n))
print([a(n) for n in range(1, 76)]) # Michael S. Branicky, Nov 26 2021

Formula

Dirichlet g.f.: ( zeta(s) * primezeta(s-1) )^2.
a(n) = Sum_{d|n} A061397(d) * A319131(n/d).
a(p) = 0 for p prime. - Michael S. Branicky, Nov 26 2021
a(p^k) = (k-1)*p^2 for p prime and k > 0. - Chai Wah Wu, Nov 28 2021

A369893 a(n) = Sum_{p|n, p prime} n^tau(n/p).

Original entry on oeis.org

0, 2, 3, 16, 5, 72, 7, 512, 81, 200, 11, 22464, 13, 392, 450, 65536, 17, 110808, 19, 168000, 882, 968, 23, 191434752, 625, 1352, 19683, 636608, 29, 2430000, 31, 33554432, 2178, 2312, 2450, 4353564672, 37, 2888, 3042, 4098560000, 41, 9335088, 43, 3833280
Offset: 1

Views

Author

Wesley Ivan Hurt, Feb 04 2024

Keywords

Crossrefs

Cf. A000005 (tau), A062799, A319131.

Programs

  • Mathematica
    Table[DivisorSum[n, n^DivisorSigma[0, n/#] &, PrimeQ[#] &], {n, 60}]

Formula

a(p^k) = p^(k^2), for prime p and k >= 1. - Wesley Ivan Hurt, Jun 26 2024

A369739 a(n) = Sum_{p|n, p prime} p^tau(n/p).

Original entry on oeis.org

0, 2, 3, 4, 5, 13, 7, 8, 9, 29, 11, 43, 13, 53, 34, 16, 17, 89, 19, 141, 58, 125, 23, 145, 25, 173, 27, 359, 29, 722, 31, 32, 130, 293, 74, 793, 37, 365, 178, 689, 41, 2498, 43, 1347, 206, 533, 47, 499, 49, 633, 298, 2213, 53, 745, 146, 2465, 370, 845, 59, 16610, 61, 965, 424, 64, 194
Offset: 1

Views

Author

Wesley Ivan Hurt, Jan 30 2024

Keywords

Crossrefs

Cf. A000005 (tau), A319131, A345270.

Programs

  • Mathematica
    Table[DivisorSum[n, #^DivisorSigma[0, n/#] &, PrimeQ[#] &], {n, 60}]

Formula

a(p^k) = p^k, for p prime and k >= 1. - Wesley Ivan Hurt, Jun 26 2024
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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