A072026 Swap twin prime pairs >(3,5) in prime factorization of n.

1, 2, 3, 4, 7, 6, 5, 8, 9, 14, 13, 12, 11, 10, 21, 16, 19, 18, 17, 28, 15, 26, 23, 24, 49, 22, 27, 20, 31, 42, 29, 32, 39, 38, 35, 36, 37, 34, 33, 56, 43, 30, 41, 52, 63, 46, 47, 48, 25, 98, 57, 44, 53, 54, 91, 40, 51, 62, 61, 84, 59, 58, 45, 64, 77, 78

Offset: 1

Author

Reinhard Zumkeller, Jun 07 2002

Keywords

• nonn
• mult

Examples

			a(143)=a(11*13)=a(11)*a(13)=13*11=143; a(77)=a(7*11)=a(7)*a(11)=5*13=65.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Index entries for sequences that are permutations of the natural numbers.

Crossrefs

Cf. A007510, A037074, A072027, A001359, A006512, A072028, A072029, A061898, A064505.

Programs

Mathematica

a[n_] := Product[{p, e} = pe; If[p <= 3, p, If[PrimeQ[p+2], p+2, If[PrimeQ[p-2], p-2, p]]]^e, {pe, FactorInteger[n]}];
Array[a, 100] (* Jean-François Alcover, Nov 20 2021 *)

Formula

a(a(n)) = n, a self-inverse permutation of natural numbers.

a(n) = n for single primes (A007510) and products of twin prime pairs (A037074).

Multiplicative with a(p) = (if p<=3 then p else (if p+2 is prime then p+2 else (if p-2 is prime then p-2 else p))), p prime.

Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p, q primes > 3, p = q+2} ((p^2-p)*(q^2-q)/((p^2-q)*(q^2-p))) = 0.53439004468579249988... . - Amiram Eldar, Dec 24 2022