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A072028 Swap twin prime pairs of form (4*k+1,4*k+3) in prime factorization of n. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
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
Sequence in context: A343600 A343601 A331275 * A269377 A072026 A332213
KEYWORD
nonn,mult
AUTHOR
Reinhard Zumkeller, Jun 07 2002
STATUS
approved

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Last modified March 28 15:38 EDT 2024. Contains 371254 sequences. (Running on oeis4.)