A072028 Swap twin prime pairs of form (4*k+1,4*k+3) in prime factorization of n.

1, 2, 3, 4, 7, 6, 5, 8, 9, 14, 11, 12, 13, 10, 21, 16, 19, 18, 17, 28, 15, 22, 23, 24, 49, 26, 27, 20, 31, 42, 29, 32, 33, 38, 35, 36, 37, 34, 39, 56, 43, 30, 41, 44, 63, 46, 47, 48, 25, 98, 57, 52, 53, 54, 77, 40, 51, 62, 59, 84, 61, 58, 45, 64, 91, 66

Offset: 1

Author

Reinhard Zumkeller, Jun 07 2002

Keywords

• nonn
• mult

Examples

			a(65) = a(5*13) = a(5)*a(13) = a(4*1+1)*a(13) = (4*1+3)*13 = 7*13 = 91.
		

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. A061898, A064505, A071695, A071696, A072026, A072027, A072029.

Programs

Mathematica

f[p_, e_] := If[p < 5, p, If[(m = Mod[p, 4]) == 1 && PrimeQ[p + 2], p + 2, If[m == 3 && PrimeQ[p - 2], p - 2, p]]]^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 26 2024 *)

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, p = f[i,1]; if(p < 5, p, if(p%4 == 1 && isprime(p+2), p+2, if(p%4 == 3 && isprime(p-2), p-2, p)))^f[i,2]);} \\ Amiram Eldar, Feb 26 2024

Formula

Multiplicative with a(p) = (if p mod 4 = 1 and p+2 is prime then p+2 else (if p mod 4 = 3 and p-2 is prime then p-2 else p)), p prime.

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

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))) = 1.0627249749498993391... . - Amiram Eldar, Feb 26 2024