A294370 Lexicographically earliest sequence of distinct positive numbers such that, for any n > 0, a(2*n) = 3*a(n).

1, 3, 2, 9, 4, 6, 5, 27, 7, 12, 8, 18, 10, 15, 11, 81, 13, 21, 14, 36, 16, 24, 17, 54, 19, 30, 20, 45, 22, 33, 23, 243, 25, 39, 26, 63, 28, 42, 29, 108, 31, 48, 32, 72, 34, 51, 35, 162, 37, 57, 38, 90, 40, 60, 41, 135, 43, 66, 44, 99, 46, 69, 47, 729, 49, 75

Offset

1,2

Comments

This sequence can be generalized easily: for any i > 1 and j > 1 such that gcd(i, j)=1:

- let f_i_j be the lexicographically earliest sequence of distinct positive numbers such that, for any n > 0, f_i_j(i*n) = j*f_i_j(n),

- in particular, f_2_3 = a (this sequence),

- if n is the k-th positive number not divisible by i and m the k-th positive number not divisible by j, then f_i_j(n) = m, and for any x >= 0, f_i_j(n*i^x) = m*j^x,

- a(n) is divisible by j^x iff n is divisible by i^x,

- f_i_j is a permutation of the natural numbers, with inverse f_j_i,

- f_i_j(1) = 1.

See A294371 for the inverse of this sequence.

Apparently, a(1) = 1 and a(6) = 6 are the only fixed points of this sequence.

Links

Table of n, a(n) for n=1..66.

Index entries for sequences that are permutations of the natural numbers

Formula

a(A005408(i)*2^j) = A001651(i)*3^j for any i > 0 and j >= 0.

Example

a(1) = 1 is suitable, and a(2^i) = 3^i for any i >= 0.
a(2) = 3 * a(1) = 3.
a(3) = 2 is suitable, and a(3*2^i) = 2*3^i for any i >= 0.
a(4) = 3 * a(2) = 9.
a(5) = 4 is suitable, and a(5*2^i) = 4*3^i for any i >= 0.

Crossrefs

Cf. A001651, A005408, A294371 (inverse).

Sequence in context: A224578 A134562 A090639 * A325984 A178774 A379658

Adjacent sequences: A294367 A294368 A294369 * A294371 A294372 A294373

Keyword

nonn,easy

Author

Rémy Sigrist, Oct 29 2017

Status

approved