A287928 - OEIS

A287928

Lexicographically earliest sequence of distinct positive terms such that, if digsum(a(i)) = digsum(a(j)), then either i = j or digsum(a(i+1)) != digsum(a(j+1)) (where digsum is the digital sum, A007953).

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 11, 13, 15, 14, 16, 18, 17, 19, 20, 23, 21, 24, 22, 25, 28, 26, 29, 27, 30, 32, 31, 35, 33, 36, 34, 38, 37, 39, 40, 45, 41, 44, 43, 42, 46, 47, 48, 49, 50, 54, 51, 53, 57, 52, 58, 55, 59, 56, 60, 65, 61, 66, 62, 67, 63, 64

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

This sequence is a permutation of the natural numbers, with inverse A287929.

More generally, if g is a function over the natural numbers with infinitely many distinct values, then there is a lexicographically earliest sequence of distinct positive terms, say f_g, such that, if g(f_g(i)) = g(f_g(j)), then either i = j or g(f_g(i+1)) != g(f_g(j+1)), and f_g is a permutation of the natural numbers:

- in particular, f_A007953 = a,

- and f_tau = A175500 (where tau = A000005),

- if g is injective then f_g = A000027.

Among the first 250000 terms, we have the following fixed points: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 16, 19, 20, 25, 30, 39, 40, 46, 47, 48, 49, 50, 53, 60, 70, 76, 79, 80, 88, 89, 90, 92, 99, 100, 108, 111, 126, 193

, 675.

LINKS

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Rémy Sigrist, PARI program for A287928

Rémy Sigrist, Logarithmic scatterplot of the first 250000 terms

Rémy Sigrist, Illustration of the first terms

Index entries for sequences that are permutations of the natural numbers

EXAMPLE

For n = 1..9, a(n) = n satisfies the definition, and digsum(a(n)) = n.

Also a(10) = 10 satisfies the definition, and digsum(a(10)) = 1.

As digsum(a(10)) = digsum(a(1)), digsum(a(11)) != digsum(a(2)).

a(11) = 12 satisfies the definition.