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

Author

Marc LeBrun, May 14 2001

Keywords

• easy
• nonn
• look
• mult

Comments

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

Examples

			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).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..20000
• Index entries for sequences that are permutations of the natural numbers.

Crossrefs

Cf. A045965, A275407 (fixed points).

Programs

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 *)

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

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