A305458 In primorial base: a(n) is obtained by replacing each nonzero digit of n with its product with the nonzero digits at lower indices (See Comments for precise definition).

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 16, 17, 12, 13, 14, 15, 28, 29, 18, 19, 20, 21, 10, 11, 24, 25, 26, 27, 22, 23, 30, 31, 32, 33, 64, 65, 36, 37, 38, 39, 76, 77, 72, 73, 74, 75, 148, 149, 108, 109, 110, 111, 190, 191, 144, 145, 146, 147, 52, 53, 60, 61, 62, 63

Offset

0,3

Comments

For a number n >= 0, let d_k, ..., d_0 be the digits of n in primorial base (n = Sum_{i=0..k} d_i * A002110(i), and for i = 0..k, 0 <= d_i < prime(i+1)); the digits of a(n) in primorial base, say e_k, ..., e_0, satisfy: for i = 0..k:

- if d_i = 0, then e_i = 0,

- if d_i > 0, then e_i == Product_{j=0..i and d_j > 0} d_j (mod prime(i+1)).

This sequence is a permutation of the nonnegative integers with inverse A305463.

The fixed points of this sequence (A305462) correspond to the numbers with all digits, except possibly the leading digit, equal to zero or one in primorial base.

Like A289234, this sequence preserves the number of digits and the number of nonzero digits in primorial base.

For any prime number p:

- we can build an analog of this sequence, say f_p, for the base p,

- in particular, f_2 = A001477,

- f_p is a permutation of the nonnegative integers,

- f_p preserves the number of digits and the number of nonzero digits in base p,

- the fixed points of f_p correspond to the numbers with all digits, except possibly the leading digit, equal to zero or one in base p.

Links

Rémy Sigrist, Table of n, a(n) for n = 0..2310

Index entries for sequences that are permutations of the natural numbers

Index entries for sequences related to primorial base

Formula

A235224(a(n)) = A235224(n).

A267263(a(n)) = A267263(n).

Example

The digits of 7772 in primorial base are 3,4,0,0,1,0.
Also:
- 1 == 1 (mod prime(2)),
- 4 * 1 == 4 (mod prime(5)),
- 3 * 4 * 1 == 12 (mod prime(6)).
Hence the digits of a(7772) in primorial base are 12,4,0,0,1,0, and a(7772) = 28562.

Prog

(PARI) a(n) = my (v=0, k=1, r=2, p=1); while (n, my (d=n % r); if (d, k *= d; v += p * lift(Mod(k, r))); n \= r; p *= r; r = nextprime(r+1)); return (v)

Crossrefs

Cf. A001477, A002110, A049345, A235224, A267263, A289234, A305462 (fixed points), A305463 (inverse).

Sequence in context: A081433 A032581 A062504 * A102489 A004838 A032998

Adjacent sequences: A305455 A305456 A305457 * A305459 A305460 A305461

Keyword

nonn,base

Author

Rémy Sigrist, Jun 01 2018

Status

approved