A250249 Permutation of natural numbers: a(1) = 1, a(n) = A083221(a(A055396(n)), a(A246277(n))).

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 27, 22, 23, 24, 25, 26, 21, 28, 29, 30, 31, 32, 39, 34, 35, 36, 37, 38, 63, 40, 41, 54, 43, 44, 33, 46, 47, 48, 49, 50, 75, 52, 53, 42, 65, 56, 99, 58, 59, 60, 61, 62, 57, 64, 95, 78, 67, 68, 111, 70, 71, 72, 103, 74, 51, 76, 77, 126, 79, 80, 45, 82

Offset

1,2

Comments

This is a "doubly-recursed" version of A249817.

For primes p_n, a(p_n) = p_{a(n)}.

The first 7-cycle occurs at: (33 39 63 57 99 81 45), which is mirrored by the cycle (66 78 126 114 198 162 90) with terms double the size and also by the cycle (137 167 307 269 523 419 197), consisting of primes (p_33, p_39, p_63, ...).

Links

Antti Karttunen, Table of n, a(n) for n = 1..3718

Index entries for sequences that are permutations of the natural numbers

Formula

a(1) = 1, a(n) = A083221(a(A055396(n)), a(A246277(n))).

Other identities. For all n >= 1:

a(2n) = 2*a(n), or equally, a(n) = a(2n)/2. [The even bisection halved gives the sequence back].

a(p_n) = p_{a(n)}, or equally, a(n) = A049084(a(A000040(n))). [Restriction to primes induces the same sequence].

A078442(a(n)) = A078442(n), A049076(a(n)) = A049076(n). [Preserves the "order of primeness of n"].

A000035(n) = A000035(a(n)). [Preserves the parity].

Example

For n = 42 = 2*3*7, we see that it occurs as the 21st term on the top row of A246278 (A055396(42) = 1 and A246277(42) = 21), recursing on both yields a(1) = 1, a(21) = 27, thus we find A083221(1,27), the 27th term on A083221's topmost row (also A005843) which is 54, thus a(42) = 54.
Examples for cases where n is a prime:
a(3709) = a(p_518) = p_{a(518)} = A000040(1162) = 9397.
a(3719) = a(p_519) = p_{a(519)} = A000040(1839) = 15767.

Prog

(Scheme, with memoizing-macro definec from Antti Karttunen's IntSeq-library)
(definec (A250249 n) (cond ((<= n 1) n) (else (A083221bi (A250249 (A055396 n)) (A250249 (A246277 n)))))) ;; Code for A083221bi given in A083221

Crossrefs

Inverse: A250250.

Fixed points: A250251, their complement: A249729.

Cf. A000035, A000040, A055396, A078442, A083221, A049076.

Differs from A250250 for the first time at n=33, where a(33) = 39, while A250250(33) = 45.

Differs from the "vanilla version" A249817 for the first time at n=42, where a(42) = 54, while A249817(42) = 42.

Sequence in context: A249817 A250247 A250245 * A249818 A250248 A250246

Adjacent sequences: A250246 A250247 A250248 * A250250 A250251 A250252

Keyword

nonn

Author

Antti Karttunen, Nov 17 2014

Status

approved