A072029 Swap twin prime pairs of form (4*k+3,4*(k+1)+1) in prime factorization of n.

1, 2, 5, 4, 3, 10, 7, 8, 25, 6, 13, 20, 11, 14, 15, 16, 17, 50, 19, 12, 35, 26, 23, 40, 9, 22, 125, 28, 29, 30, 31, 32, 65, 34, 21, 100, 37, 38, 55, 24, 41, 70, 43, 52, 75, 46, 47, 80, 49, 18, 85, 44, 53, 250, 39, 56, 95, 58, 61, 60, 59, 62, 175, 64, 33

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 mod 4 = 3 and p+2 is prime then p+2 else (if p mod 4 = 1 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))) = 1.37140598833326962... . - Amiram Eldar, Feb 26 2024

Example

a(42) = a(2*3*7) = a(2)*a(3)*a(7) = a(2)*a(4*0+3)*a(7) = 2*(4*1+1)*7 = 2*5*7 = 70.

Mathematica

a[n_] := Product[{p, e} = pe; Which[
     Mod[p, 4] == 3 && PrimeQ[p + 2], p + 2,
     Mod[p, 4] == 1 && PrimeQ[p - 2], p - 2,
     True, p]^e, {pe, FactorInteger[n]}];
Array[a, 100] (* Jean-François Alcover, Nov 21 2021 *)

Prog

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

Crossrefs

Cf. A061898, A064505, A071698, A071699, A072026, A072027, A072028.

Sequence in context: A266403 A266415 A132664 * A309734 A309668 A238758

Adjacent sequences: A072026 A072027 A072028 * A072030 A072031 A072032

Keyword

nonn,mult

Author

Reinhard Zumkeller, Jun 07 2002

Status

approved