A092520 -id:A092520 - cp's OEIS frontend

A226602 Number of ordered triples (i,j,k) with ijk = n, i,j,k >= 0 and gcd(i,j,k) <= 1.

1, 1, 3, 3, 6, 3, 9, 3, 9, 6, 9, 3, 18, 3, 9, 9, 12, 3, 18, 3, 18, 9, 9, 3, 27, 6, 9, 9, 18, 3, 27, 3, 15, 9, 9, 9, 36, 3, 9, 9, 27, 3, 27, 3, 18, 18, 9, 3, 36, 6, 18, 9, 18, 3, 27, 9, 27, 9, 9, 3, 54, 3, 9, 18, 18, 9, 27, 3, 18, 9, 27, 3, 54, 3, 9, 18, 18

Offset: 0

Robert Price, Jun 13 2013

Note that gcd(0,m) = m for any m.

a(n) is the number of cubefree divisors summed over the divisors of n. In other words, a(n) = Sum_{d|n} A073184(d). - Geoffrey Critzer, Mar 20 2015

Alois P. Heinz, Table of n, a(n) for n = 0..20000 (first 101 terms from Robert Price)

Cf. A000005, A007427, A008836, A034444, A056624, A073184, A092520, A100450, A212793, A226357, A226359.

with(numtheory):
b:= proc(n, t, g) option remember; `if`(t=0,
      `if`(igcd(n, g)=1, 1, 0), add(b(n/d, t-1,
      igcd(g, d)), d=divisors(n)))
    end:
a:= n-> `if`(n=0, 1, b(n, 2, 0)):
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 20 2015

f[n_] := Length[Complement[Union[Flatten[Table[If[i*j*k == n && GCD[i, j, k] <= 1, {i, j, k}], {i, 0,n}, {j, 0, n}, {k, 0, n}], 2]], {Null}]]; Table[f[n], {n, 0, 100}]
a[0] = a[1] = 1; a[n_] := Times @@ (3 * Last[#] & /@ FactorInteger[n]); Array[a, 100, 0] (* Amiram Eldar, Sep 14 2020 *)

from math import prod
from sympy import factorint
def A226602(n): return prod(3*e for e in factorint(n).values()) if n else 1 # Chai Wah Wu, Dec 26 2022

From Geoffrey Critzer, Mar 20 2015: (Start)
If n = p_1^e_1*p_2^e_2*...*p_r^e_r then a(n) = Product_{i=1..r} 3*e_i.
Dirichlet g.f.: zeta(s)^3/zeta(3*s). (End)
From Werner Schulte, May 13 2018: (Start)
Multiplicative with a(p^e) = 3*e, p prime and e>0.
Dirichlet inverse b(n), n>0, is multiplicative with b(1) = 1, and for p prime and e>0: b(p^e)=0 if e mod 3 = 0 otherwise b(p^e)=3*(-1)^(e mod 3).
Dirichlet convolution with A007427(n) yields A212793(n).
Dirichlet convolution with A008836(n) yields A092520(n).
Equals Dirichlet convolution of A034444(n) and A056624(n).
Equals Dirichlet convolution of A000005(n) and A212793(n). (End)
Sum_{k=1..n} a(k) ~ n/(2*Zeta(3)) * (log(n)^2 + 2*log(n) * (-1 + 3*gamma - 3*Zeta'(3)/Zeta(3)) + 2 + 6*gamma^2 - 6*sg1 + 6*Zeta'(3)/Zeta(3) + 18*Zeta'(3)^2/Zeta(3)^2 - 6*gamma*(1 + 3*Zeta'(3)/Zeta(3)) - 9*Zeta''(3)/Zeta(3)), where gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). - Vaclav Kotesovec, Feb 07 2019
a(n) = A005361(n) * A074816(n). - Vaclav Kotesovec, Feb 27 2023

A318498 Denominators of the sequence whose Dirichlet convolution with itself yields A061389, number of (1+phi)-divisors of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 8, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 8, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 8, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 16, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 8, 8, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 8, 1, 2, 2, 4, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Aug 30 2018

The sequence seems to give the denominators of a few other similarly constructed rational valued sequences obtained as "Dirichlet Square Roots" (of possibly A092520 and A293443).

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A061389, A318497 (numerators), A318499.

Cf. also A299150, A046644.

up_to = 65537;
A061389(n) = factorback(apply(e -> (1+eulerphi(e)),factor(n)[,2]));
DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&dA317937.
v318497_98 = DirSqrt(vector(up_to, n, A061389(n)));
A318497(n) = numerator(v318497_98[n]);
A318498(n) = denominator(v318497_98[n]);

a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A061389(n) - Sum_{d|n, d>1, d 1.

a(n) = 2^A318499(n).

A303915 a(n) = lambda(n)*E(n), where lambda(n) = A008836(n) and E(n) = A005361(n).

Original entry on oeis.org

1, -1, -1, 2, -1, 1, -1, -3, 2, 1, -1, -2, -1, 1, 1, 4, -1, -2, -1, -2, 1, 1, -1, 3, 2, 1, -3, -2, -1, -1, -1, -5, 1, 1, 1, 4, -1, 1, 1, 3, -1, -1, -1, -2, -2, 1, -1, -4, 2, -2, 1, -2, -1, 3, 1, 3, 1, 1, -1, 2, -1, 1, -2, 6, 1, -1, -1, -2, 1, -1, -1, -6, -1, 1, -2, -2, 1, -1, -1, -4, 4, 1, -1

Offset: 1

Werner Schulte, May 02 2018

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms)

Signed version of A005361.

Cf. A008836, A048691, A092520.

Array[LiouvilleLambda[#] Apply[Times, FactorInteger[#][[All, -1]] ] &, 83] (* Michael De Vlieger, May 06 2018 *)

a(n) = (-1)^bigomega(n)*factorback(factor(n)[, 2]); \\ Michel Marcus, May 05 2018

Multiplicative with a(p^e) = e*(-1)^e, p prime and e > 0.
Dirichlet g.f.: (zeta(2*s))^2 / (zeta(s)*zeta(3*s)).
Dirichlet convolution with A048691(n) yields A092520(n).
Dirichlet inverse b(n), n>=1, is multiplicative with b(1)=1 and for p prime and e>0: b(p^e) = 0 if e mod 3 = 0 otherwise b(p^e) = (-1)^(3 - e mod 3).

A377141 Numbers whose cubes have more square divisors than the cube of any smaller number.

Original entry on oeis.org

1, 2, 4, 8, 12, 24, 36, 72, 144, 180, 360, 720, 900, 1800, 3600, 6300, 10800, 12600, 25200, 44100, 75600, 88200, 176400, 352800, 529200, 705600, 1411200, 1587600, 1940400, 3880800, 5821200, 7761600, 15523200, 17463600, 21344400, 42688800, 64033200, 85377600, 170755200

Offset: 1

Amiram Eldar, Oct 17 2024

nonn

Indices of records in A092520.

The corresponding record values are 1, 2, 4, 5, 8, 10, 16, 20, 28, 32, 40, 56, 64, 80, 112, 128, ... (see the link for more values).

Amiram Eldar, Table of n, a(n) for n = 1..355
Amiram Eldar, Table of n, a(n), A363825(a(n)) for n = 1..355

Subsequence of A025487.

Cf. A092520.

f[p_, e_] := Floor[(3*e+2)/2]; d[1] = 1; d[n_] := Times @@ f @@@ FactorInteger[n]; With[{v = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]]}, seq = {}; dm = 0; Do[If[(dk = d[v[[k]]]) > dm, dm = dk; AppendTo[seq, v[[k]]]], {k, 1, Length[v]}]; seq]

cp's OEIS Frontend

A226602 Number of ordered triples (i,j,k) with i*j*k = n, i,j,k >= 0 and gcd(i,j,k) <= 1.

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A318498 Denominators of the sequence whose Dirichlet convolution with itself yields A061389, number of (1+phi)-divisors of n.

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A303915 a(n) = lambda(n)*E(n), where lambda(n) = A008836(n) and E(n) = A005361(n).

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A377141 Numbers whose cubes have more square divisors than the cube of any smaller number.

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Mathematica

A226602 Number of ordered triples (i,j,k) with ijk = n, i,j,k >= 0 and gcd(i,j,k) <= 1.