A289234 In primorial base: a(n) is obtained by replacing each nonzero digit of n with its inverse (see Comments for precise definition).

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 18, 19, 20, 21, 22, 23, 12, 13, 14, 15, 16, 17, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 48, 49, 50, 51, 52, 53, 42, 43, 44, 45, 46, 47, 54, 55, 56, 57, 58, 59, 120, 121, 122, 123, 124, 125

Offset: 0

Author

Rémy Sigrist, Jun 28 2017

Keywords

• nonn
• base
• look

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 is the inverse of d_i mod prime(i+1) (in other words, 1 <= e_i < prime(i+1) and e_i * d_i = 1 mod prime(i+1)).

This sequence is a self-inverse permutation of the nonnegative numbers.

a(n) < A002110(k) iff n < A002110(k) for any n >= 0 and k >= 0.

a(n) = n iff the digits of n in primorial base, say d_k, ..., d_0, satisfy: for i=0..k: d_i = 0, 1 or prime(i+1)-1.

For k > 0: the plotting of the first A002110(k) terms can be obtained by arranging prime(k) copies of the plotting of the first A002110(k-1) terms in a prime(k) X prime(k) grid:

- one copy in the cell at position (0,0),

- one copy in any cell at position (i,j) with i*j = 1 mod prime(k) (with 0 < i < prime(k) and 0 < j < prime(k)).

Examples

			The digits of 7772 in primorial base are 3,4,0,0,1,0.
Also:
- 9 * 3 = 27 = 1 mod prime(6) = 13,
- 3 * 4 = 12 = 1 mod prime(5) = 11,
- 1 * 1 = 1 mod prime(2) = 3.
Hence, the digits of a(7772) in primorial base are 9,3,0,0,1,0,
and a(7772) = 9 * 11# + 3 * 7# + 1 * 2# = 21422.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..30029
• Rémy Sigrist, Scatterplot of the first 510510 (=17#) terms
• Index entries for sequences that are permutations of the natural numbers
• Index entries for sequences related to primorial base

Crossrefs

Cf. A002110, A049345, A276085, A276086, A328617.

Cf. also A231716, A328622, A328623, A328625, A328626.

Programs

PARI

a(n) = my (pr=1, p=2, v=0); while (n>0, my (d=n%p); if (d>0, v += pr * lift(1/Mod(d,p))); pr *= p; n \= p; p = next prime(p+1)); return (v)

Formula

a(n) = A276085(A328617(A276086(n))). - Antti Karttunen, Oct 26 2019