A061898 Swap each prime in factorization of n with "neighbor" prime.

1, 3, 2, 9, 7, 6, 5, 27, 4, 21, 13, 18, 11, 15, 14, 81, 19, 12, 17, 63, 10, 39, 29, 54, 49, 33, 8, 45, 23, 42, 37, 243, 26, 57, 35, 36, 31, 51, 22, 189, 43, 30, 41, 117, 28, 87, 53, 162, 25, 147, 38, 99, 47, 24, 91, 135, 34, 69, 61, 126, 59, 111, 20, 729, 77, 78, 71, 171, 58

Offset

1,2

Comments

Here "neighbor" primes are just paired in order: 2<->3, 5<->7, 11<->13, etc. Self-inverse permutation of the integers. Multiplicative.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Index entries for sequences that are permutations of the natural numbers.

Formula

Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((p^2-p)/(p^2-q(p)) = 0.9229142333..., where q(p) is the "neighbor" of p. - Amiram Eldar, Nov 29 2022

Example

a(60) = 126 since 60 = 2^2*3*5, swapping 2<->3 and 5<->7 gives 3^2*2*7 = 126 (and of course then a(126) = 60).

Maple

p:= proc(n) option remember; `if`(numtheory[pi](n)::odd,
       nextprime(n), prevprime(n))
    end:
a:= n-> mul(p(i[1])^i[2], i=ifactors(n)[2]):
seq(a(n), n=1..80);  # Alois P. Heinz, Sep 13 2017

Mathematica

p[n_] := p[n] = If[OddQ[PrimePi[n]], NextPrime[n], NextPrime[n, -1]];
a[1] = 1; a[n_] := Product[p[i[[1]]]^i[[2]], {i, FactorInteger[n]}];
Array[a, 80] (* Jean-François Alcover, Jun 10 2018, after Alois P. Heinz *)

Prog

(PARI) a(n) = my(f=factor(n)); for (i=1, #f~, ip = primepi(f[i, 1]); if (ip % 2, f[i, 1] = prime(ip+1), f[i, 1] = prime(ip-1))); factorback(f); \\ Michel Marcus, Jun 09 2014

Crossrefs

Cf. A045965, A275407 (fixed points).

Sequence in context: A286251 A361518 A072027 * A360648 A359036 A021756

Adjacent sequences: A061895 A061896 A061897 * A061899 A061900 A061901

Keyword

easy,nonn,look,mult

Author

Marc LeBrun, May 14 2001

Status

approved