A306458 -id:A306458 - cp's OEIS frontend

A363194 Number of divisors of the n-th powerful number A001694(n).

1, 3, 4, 3, 5, 3, 4, 6, 9, 3, 7, 12, 5, 9, 12, 3, 4, 8, 15, 3, 9, 12, 16, 9, 6, 9, 18, 3, 15, 4, 3, 12, 15, 20, 9, 9, 12, 10, 3, 21, 5, 20, 12, 9, 7, 15, 18, 3, 24, 27, 3, 12, 18, 16, 11, 9, 12, 24, 9, 9, 25, 12, 4, 12, 3, 12, 9, 9, 18, 21, 3, 28, 27, 36, 3, 15

Offset: 1

Amiram Eldar, May 21 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk and Matilde Lalín, Asymptotics of sums of divisor functions over sequences with restricted factorization structure, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634, eq. (8).

Cf. A000005, A001694, A180114, A306458, A343443.

Similar sequences: A072048, A076400, A363195.

DivisorSigma[0, Select[Range[3000], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &]]

apply(numdiv, select(ispowerful, [1..3000]))

from itertools import count, islice
from math import prod
from sympy import factorint
def A363194_gen(): # generator of terms
    for n in count(1):
        f = factorint(n).values()
        if all(e>1 for e in f):
            yield prod(e+1 for e in f)
A363194_list = list(islice(A363194_gen(),20)) # Chai Wah Wu, May 21 2023

a(n) = A000005(A001694(n)).
Sum_{A001694(k) < x} a(k) = c_1 * sqrt(x) * log(x)^2 + c_2 * sqrt(x) * log(x) + c_3 * sqrt(x) + O(x^(5/12 + eps)), where c_1, c_2 and c_3 are constants. c_1 = Product_{p prime} (1 + 4/p^(3/2) - 1/p^2 - 6/p^(5/2) + 2/p^(7/2))/8 = 0.516273682988566836609... . [corrected Sep 21 2024]
a(n) = A343443(A306458(n)). - Amiram Eldar, Sep 01 2023

A348018 a(n) is the index of A064549(n) = n * Product_{p prime|n} p in the sequence of powerful numbers (A001694).

Original entry on oeis.org

1, 2, 4, 3, 6, 9, 10, 5, 7, 14, 16, 12, 20, 21, 24, 8, 28, 15, 31, 22, 35, 36, 39, 19, 17, 44, 13, 32, 48, 50, 51, 11, 56, 59, 60, 23, 65, 67, 68, 33, 71, 73, 75, 52, 43, 81, 84, 27, 30, 37, 90, 64, 94, 29, 97, 46, 102, 104, 107, 74, 110, 111, 62, 18, 117, 119

Offset: 1

Amiram Eldar, Sep 24 2021

nonn

A permutation of the positive integers.

The inverse permutation of A306458.

			The sequence of powerful numbers (A001694) begins with 1, 4, 8, 9, ...
The position of A064549(1) = 1 in A001694 is 1, so a(1) = 1.
The position of A064549(3) = 9 in A001694 is 4, so a(3) = 4.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001694, A064549, A306458.

powQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], # > 1 &]; pows = Join[{1}, Select[Range[10^4], powQ]]; TakeWhile[Table[FirstPosition[pows, n * Times @@ (First /@ FactorInteger[n])][[1]], {n, 1, 100}], NumericQ]

A001694(a(n)) = A064549(n).
A306458(a(n)) = a(A306458(n)) = n.
The fixed points of this permutation are 1, 2, 12, 1208, 1256, 1288 and no more below 3*10^5.

A370328 The number of powerful divisors of the powerful numbers.

Original entry on oeis.org

1, 2, 3, 2, 4, 2, 3, 5, 4, 2, 6, 6, 4, 4, 6, 2, 3, 7, 8, 2, 4, 6, 9, 4, 5, 8, 10, 2, 8, 3, 2, 6, 8, 12, 4, 4, 6, 9, 2, 12, 4, 12, 6, 4, 6, 8, 10, 2, 15, 8, 2, 6, 10, 9, 10, 4, 6, 14, 4, 4, 16, 6, 3, 6, 2, 6, 4, 4, 10, 12, 2, 18, 8, 12, 2, 8, 15, 12, 8, 11, 4, 7

Offset: 1

Amiram Eldar, Feb 15 2024

The product of the exponents of the prime factorization of the powerful numbers.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A001694, A005361, A306458, A363194, A370329.

f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[n == 1 || Min[e] > 1, Times @@ e, Nothing]]; Array[f, 2500]

lista(kmax) = {my(e); for(k = 1, kmax, e = factor(k)[,2]; if(k == 1 || vecmin(e) > 1, print1(vecprod(e), ", ")));}

a(n) = A005361(A001694(n)).

a(n) = A000005(A306458(n)).

cp's OEIS Frontend

A363194 Number of divisors of the n-th powerful number A001694(n).

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A348018 a(n) is the index of A064549(n) = n * Product_{p prime|n} p in the sequence of powerful numbers (A001694).

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Examples

Links

Crossrefs

Programs

Mathematica

Formula

A370328 The number of powerful divisors of the powerful numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula