A235200 Self-inverse and multiplicative permutation of integers: a(0)=0, a(1)=1, a(2)=2, a(3)=5 and a(5)=3, a(p_i) = p_{a(i-1)+1} for primes with index i > 3, and a(u * v) = a(u) * a(v) for u, v > 0.

0, 1, 2, 5, 4, 3, 10, 13, 8, 25, 6, 11, 20, 7, 26, 15, 16, 31, 50, 43, 12, 65, 22, 23, 40, 9, 14, 125, 52, 101, 30, 17, 32, 55, 62, 39, 100, 37, 86, 35, 24, 73, 130, 19, 44, 75, 46, 103, 80, 169, 18, 155, 28, 53, 250, 33, 104, 215, 202, 59, 60, 131, 34, 325, 64

Offset

0,3

Comments

The permutation satisfies A065091(a(n)) = a(A065091(n)) for all n >= 3, and is self-inverse: It swaps 3 & 5, maps all larger primes p_i (with index i > 3) to p_{a(i-1)+1}, and lets the multiplicativity take care of the rest.

Links

Antti Karttunen, Table of n, a(n) for n = 0..5520

Index entries for sequences that are permutations of the natural numbers

Formula

For n < 3, a(n)=n, a(3)=5 and a(5)=3, a(p_i) = p_{a(i-1)+1} for primes with index i > 3, and a(u * v) = a(u) * a(v) for u, v > 0.

A000035(a(n)) = A000035(n) = (n mod 2) for all n. [Even terms occur only on even indices and odd terms only on odd indices, respectively]

Prog

(Scheme, with Antti Karttunen's IntSeq-library)
(definec (A235200 n) (cond ((< n 2) n) ((= n 3) 5) ((= n 5) 3) ((= 1 (A010051 n)) (A000040 (+ 1 (A235200 (- (A000720 n) 1))))) (else (reduce * 1 (map A235200 (ifactor n))))))

Crossrefs

List below gives similarly constructed permutations, which all force a swap of two small numbers, with the composite numbers defined by the multiplicative property and the fact that (the rest of) primes are permuted with the sequence itself. Apart from the first one, all satisfy A000040(a(n)) = a(A000040(n)) except for a finite number of cases (with A234840, substitute A008578 for A000040):

A234840 (swaps 2 & 3).

A235199 (swaps 5 & 7).

A235201 (swaps 3 & 4).

A235487 (swaps 7 & 8).

A235489 (swaps 8 & 9).

Cf. also A072029, A181351, A234743/A234744.

Sequence in context: A238758 A354182 A065652 * A267099 A266416 A266401

Adjacent sequences: A235197 A235198 A235199 * A235201 A235202 A235203

Keyword

nonn,mult

Author

Antti Karttunen, Jan 04 2014

Status

approved