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

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

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

Example

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

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 *)

Crossrefs

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

Sequence in context: A331275 A072028 A269377 * A332213 A235199 A270426

Adjacent sequences: A072023 A072024 A072025 * A072027 A072028 A072029

Keyword

nonn,mult

Author

Reinhard Zumkeller, Jun 07 2002

Status

approved