A072027 Swap (2,3) and all twin prime pairs >(3,5) in prime factorization of n.

1, 3, 2, 9, 7, 6, 5, 27, 4, 21, 13, 18, 11, 15, 14, 81, 19, 12, 17, 63, 10, 39, 23, 54, 49, 33, 8, 45, 31, 42, 29, 243, 26, 57, 35, 36, 37, 51, 22, 189, 43, 30, 41, 117, 28, 69, 47, 162, 25, 147, 38, 99, 53, 24, 91, 135, 34, 93, 61, 126, 59, 87, 20, 729, 77

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

Multiplicative with a(p) = (if p<=3 then 5-p else (if p+2 is prime then p+2 else (if p-2 is prime then p-2 else p))), p prime.

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

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

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{(p < q) swapped pair} ((p^2-p)*(q^2-q)/((p^2-q)*(q^2-p))) = 1.832194438922717... . - Amiram Eldar, Feb 26 2024

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

f[p_, e_] := If[p < 5, 5 - p, If[PrimeQ[p + 2], p + 2, If[PrimeQ[p - 2], p - 2, p]]]^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 26 2024 *)

Prog

(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, p = f[i, 1]; if(p < 5, 5-p, if(isprime(p+2), p+2, if(isprime(p-2), p-2, p)))^f[i, 2]); } \\ Amiram Eldar, Feb 26 2024

Crossrefs

Cf. A001359, A006512, A007510, A037074, A061898, A064505, A072026, A072028, A072029.

Sequence in context: A227630 A286251 A361518 * A061898 A360648 A359036

Adjacent sequences: A072024 A072025 A072026 * A072028 A072029 A072030

Keyword

nonn,mult

Author

Reinhard Zumkeller, Jun 07 2002

Status

approved