A072028 - OEIS

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

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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 = 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

EXAMPLE

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

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 *)

PROG

(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

CROSSREFS

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

Sequence in context: A343600 A343601 A331275 * A269377 A072026 A332213

Adjacent sequences: A072025 A072026 A072027 * A072029 A072030 A072031

KEYWORD

nonn,mult

AUTHOR

Reinhard Zumkeller, Jun 07 2002

STATUS

approved