A253047 Start with the natural numbers 1,2,3,...; interchange 2*prime(i) and 3*prime(i+1) for each i, and interchange prime(prime(i)) with prime(2*prime(i)) for each i.

1, 2, 7, 9, 13, 15, 3, 8, 4, 21, 29, 12, 5, 33, 6, 16, 43, 18, 19, 20, 10, 39, 23, 24, 25, 51, 27, 28, 11, 30, 79, 32, 14, 57, 35, 36, 37, 69, 22, 40, 101, 42, 17, 44, 45, 87, 47, 48, 49, 50, 26, 52, 53, 54, 55, 56, 34, 93, 139, 60, 61, 111, 63

Offset: 1

Author

N. J. A. Sloane, Dec 26 2014

Keywords

• nonn

Comments

This is an involution on the natural numbers: applying it twice gives the identity permutation.

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000
• A. B. Frizell, Certain non-enumerable sets of infinite permutations. Bull. Amer. Math. Soc. 21 (1915), no. 10, 495-499.
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A064614, A253046, A253046.

Programs

Maple

f:= proc(t)
local r;
  if t mod 2 = 0 and isprime(t/2) then  3*nextprime(t/2)
  elif t mod 3 = 0 and isprime(t/3) then 2*prevprime(t/3)
  elif isprime(t) then
    r:= numtheory:-pi(t);
    if isprime(r) then ithprime(2*r)
    elif r mod 2 = 0 and isprime(r/2) then ithprime(r/2)
    else t
    fi
  else t
  fi
end proc:
seq(f(i),i=1..100); # Robert Israel, Dec 26 2014

Mathematica

f[t_] := Module[{r}, Which[EvenQ[t] && PrimeQ[t/2], 3 NextPrime[t/2], Divisible[t, 3] && PrimeQ[t/3], 2 NextPrime[t/3, -1], PrimeQ[t], r = PrimePi[t]; Which[PrimeQ[r], Prime[2r], EvenQ[r] && PrimeQ[r/2], Prime[r/2], True, t], True, t]];
Array[f, 100] (* Jean-François Alcover, Jul 27 2020, after Robert Israel *)

Python

from sympy import isprime, primepi, prevprime, nextprime, prime
def A253047(n):
    if n <= 2:
        return n
    if n == 3:
        return 7
    q2, r2 = divmod(n,2)
    if r2:
        q3, r3 = divmod(n,3)
        if r3:
            if isprime(n):
                m = primepi(n)
                if isprime(m):
                    return prime(2*m)
                x, y = divmod(m,2)
                if not y:
                    if isprime(x):
                        return prime(x)
            return n
        if isprime(q3):
            return 2*prevprime(q3)
        return n
    if isprime(q2):
        return 3*nextprime(q2)
    return n # Chai Wah Wu, Dec 27 2014

Extensions

Definition supplied by Robert Israel, Dec 26 2014

Offset changed to 1 by Chai Wah Wu, Dec 27 2014