A257725 Permutation of natural numbers: a(0) = 0, a(lucky(n)) = 1 + 2*a(n-1), a(unlucky(n)) = 2*a(n), where lucky(n) = n-th lucky number A000959, unlucky(n) = n-th unlucky number A050505.

0, 1, 2, 3, 4, 6, 8, 5, 12, 7, 16, 10, 24, 9, 14, 13, 32, 20, 48, 18, 28, 17, 26, 64, 40, 11, 96, 36, 56, 34, 52, 25, 128, 15, 80, 22, 192, 33, 72, 112, 68, 104, 50, 21, 256, 30, 160, 44, 384, 49, 66, 19, 144, 224, 136, 208, 100, 42, 512, 60, 320, 88, 768, 29, 98, 132, 38, 27, 288, 65, 448, 272, 416, 41, 200, 97, 84, 1024, 120, 37

Offset: 0

Author

Antti Karttunen, May 06 2015

Keywords

• nonn

Comments

In other words, after a(0) = 0, if n is the k-th lucky number [i.e., n = A000959(k)], a(n) = 1 + 2*a(k-1); otherwise, when n is the k-th unlucky number [i.e., n = A050505(k)], a(n) = 2*a(k).

Because all lucky numbers are odd, it means that odd numbers occur in odd positions only (together with some even numbers, for each one of which there is a separate infinite cycle), while the even positions contain only even numbers.

Links

• Antti Karttunen, Table of n, a(n) for n = 0..10000
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Inverse: A257726.

Cf. A005408, A005843, A000959, A050505, A109497, A145649.

Related or similar permutations: A237427, A246377, A257732, A257734.

Cf. also A257690 (another similar permutation, but with a slightly different definition, resulting the first differing term at n=13, where a(13) = 9, while A257690(13) = 11).

Cf. also A257735 - A257738.

Formula

a(0) = 0; for n >= 1: if A145649(n) = 1 [i.e., if n is lucky], then a(n) = 1+(2*a(A109497(n)-1)), otherwise a(n) = 2*a(n-A109497(n)). [Where A109497(n) gives the number of lucky numbers <= n.]

As a composition of other permutations. For all n >= 1:

a(n) = A246377(A257732(n)).

a(n) = A237427(A257734(n)).

Extensions

Formula in name corrected by Antti Karttunen, Jan 10 2016