A300948 -id:A300948 - cp's OEIS frontend

A300950 Fixed points of A300948.

3, 4, 7, 8, 11, 12, 15, 48, 51, 52, 55, 56, 59, 60, 63, 64, 67, 68, 71, 72, 75, 76, 79, 112, 115, 116, 119, 120, 123, 124, 127, 128, 131, 132, 135, 136, 139, 140, 143, 176, 179, 180, 183, 184, 187, 188, 191, 192, 195, 196, 199, 200, 203, 204, 207, 240, 243

Offset: 1

Rémy Sigrist, Mar 16 2018

This sequence contains A007013(k) for any k > 0.
We can devise a set of primitive fixed points of A300948, say P, as follows:
- P contains the powers of 2, say 2^i, such that A300948(2^i) = 2^i (in that case, i = a(k)-1 for some k > 0),
- and the sums of two distinct powers of 2, say 2^i + 2^j, such that A300948(2^i) = 2^j,
- we can uniquely write any term of this sequence as a sum of distinct terms of P.

			A300948(7) = 7 hence 7 belongs to this sequence.
A300948(42) = 25 hence 42 does not belong to this sequence.

Cf. A007013, A300948.

For any n > 0 with binary expansion Sum_{i=0..k} b_i * 2^i, a(n) = Sum_{i=0..k} b_i * p(i+1) (where p(i) denotes the i-th term of the set P described in the Comments section).

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

A300950 Fixed points of A300948.

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

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