A300957 -id:A300957 - cp's OEIS frontend

A276378 Numbers k such that 6*k is squarefree.

1, 5, 7, 11, 13, 17, 19, 23, 29, 31, 35, 37, 41, 43, 47, 53, 55, 59, 61, 65, 67, 71, 73, 77, 79, 83, 85, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 127, 131, 133, 137, 139, 143, 145, 149, 151, 155, 157, 161, 163, 167, 173, 179, 181, 185, 187, 191, 193, 197, 199, 203, 205, 209, 211

Offset: 1

Juri-Stepan Gerasimov, Sep 02 2016

These are the numbers from A005117 that are not divisible by 2 and 3.
Squarefree numbers coprime to 6. - Robert Israel, Sep 02 2016
Numbers k such that A008588(k) is in A005117. - Felix Fröhlich, Sep 02 2016
The asymptotic density of this sequence is 3/Pi^2 (A104141). - Amiram Eldar, May 22 2020
From Peter Munn, Nov 20 2020: (Start)
The products generated from each subset of A215848 (primes greater than 3).
Closed under the commutative binary operation A059897(.,.), forming a subgroup of the positive integers under A059897. (End)
Multiplied by 6 we have 6, 30, 42, 66, 78, 102, ..., the values that may appear in A076978 after the 1, 2. [Don Reble, Dec 02 2020] - R. J. Mathar, Dec 15 2020
By the von Staudt-Clausen theorem, denominators of Bernoulli numbers are of the form 6*a(n) for some n. - Charles R Greathouse IV, May 16 2024

			5 is in this sequence because 6*5 = 30 = 2*3*5 is squarefree.

Robert Israel, Table of n, a(n) for n = 1..10000

Numbers m such that k*m is squarefree: A005117 (k = 1), A056911 (k = 2), A261034 (k = 3), A274546 (k = 5).
Subsequence of A007310, A300957, and A339690.
Cf. A003961, A008588, A059897, A076978, A104141, A215848.

[n: n in [1..230] | IsSquarefree(6*n)];

select(numtheory:-issqrfree, [seq(seq(6*i+j,j=[1,5]),i=0..100)]); # Robert Israel, Sep 02 2016

Select[Range@ 212, SquareFreeQ[6 #] &] (* Michael De Vlieger, Sep 02 2016 *)

is(n) = issquarefree(6*n) \\ Felix Fröhlich, Sep 02 2016

{a(n) : n >= 1} = {A003961(A003961(A005117(n))) : n >= 1} = {A003961(A056911(n)) : n >= 1}. - Peter Munn, Nov 20 2020

Sum_{n>=1} 1/a(n)^s = (6^s)*zeta(s)/((1+2^s)*(1+3^s)*zeta(2*s)), s>1. - Amiram Eldar, Sep 26 2023

A300955 In the prime tower factorization of n, replace 2's with 3's and 3's with 2's.

Original entry on oeis.org

1, 3, 2, 27, 5, 6, 7, 9, 8, 15, 11, 54, 13, 21, 10, 7625597484987, 17, 24, 19, 135, 14, 33, 23, 18, 125, 39, 4, 189, 29, 30, 31, 243, 22, 51, 35, 216, 37, 57, 26, 45, 41, 42, 43, 297, 40, 69, 47, 15251194969974, 343, 375, 34, 351, 53, 12, 55, 63, 38, 87, 59

Offset: 1

Rémy Sigrist, Mar 17 2018

The prime tower factorization of a number is defined in A182318.
This sequence is a self-inverse multiplicative permutation of the natural numbers.
This sequence has infinitely many fixed points (A300957); for any k > 0, at least one of k or 2^k * 3^a(k) is a fixed point.
This sequence is a recursive version of A182318.
This sequence has connections with A300948.

			a(6) = a(2 * 3) = 3 * 2 = 6.
a(16) = a(2 ^ 2 ^ 2) = 3 ^ 3 ^ 3 = 7625597484987.

Cf. A064614, A182318, A300948, A300957 (fixed points).

a:= n-> `if`(n=1, 1, mul(`if`(i[1]=2, 3, `if`(i[1]=3,
             2, i[1]))^a(i[2]), i=ifactors(n)[2])):
seq(a(n), n=1..60);  # Alois P. Heinz, Mar 17 2018

a[n_] := If[n == 1, 1, Product[If[i[[1]] == 2, 3, If[i[[1]] == 3,
   2, i[[1]]]]^a[i[[2]]], {i, FactorInteger[n]}]];
Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Jan 30 2025, after Alois P. Heinz *)

a(n) = my (f=factor(n)); prod(i=1, #f~, my (p=f[i,1]); if (p==2, 3, p==3, 2, p)^a(f[i,2]))

Multiplicative with a(p^k) = A064614(p)^a(k).

a(a(n)) = n.

cp's OEIS Frontend

A276378 Numbers k such that 6*k is squarefree.

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