A253046 An involution of the natural numbers: if n = 2*p_i then replace n with 3*p_{i+1}, and conversely if n = 3*p_i then replace n with 2*p_{i-1}, where p_i denotes the i-th prime.

1, 2, 3, 9, 5, 15, 7, 8, 4, 21, 11, 12, 13, 33, 6, 16, 17, 18, 19, 20, 10, 39, 23, 24, 25, 51, 27, 28, 29, 30, 31, 32, 14, 57, 35, 36, 37, 69, 22, 40, 41, 42, 43, 44, 45, 87, 47, 48, 49, 50, 26, 52, 53, 54, 55, 56, 34, 93, 59, 60, 61, 111, 63, 64, 65, 66, 67

Offset: 1

Author

N. J. A. Sloane, Dec 26 2014

Keywords

• nonn

Comments

a(m) != m iff m is a term of A253106, i.e., a semiprime divisible by 2 or 3; a(A100484(n)) > A100484(n); a(A001748(n)) < A001748(n). - Reinhard Zumkeller, Dec 26 2014

Links

• Reinhard Zumkeller, 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, A251561, A020639, A049084, A000040, A253106, A001748, A100484.

Programs

Haskell

a253046 n | i == 0 || p > 3 = n
          | p == 2          = 3 * a000040 (i + 1)
          | otherwise       = 2 * a000040 (i - 1)
            where i = a049084 (div n p);  p = a020639 n
-- Reinhard Zumkeller, Dec 26 2014

Mathematica

a253046[n_] := Block[{f},
  f[x_] := Which[PrimeQ[x/2], 3 NextPrime[x/2],
    PrimeQ[x/3], 2 NextPrime[x/3, -1],
True, x];Array[f, n]]; a253046[67] (* Michael De Vlieger, Dec 27 2014 *)

Python

from sympy import isprime, nextprime, prevprime
def A253046(n):
    q2, r2 = divmod(n,2)
    if not r2 and isprime(q2):
        return 3*nextprime(q2)
    else:
        q3, r3 = divmod(n,3)
        if not r3 and isprime(q3):
            return 2*prevprime(q3)
        return n # Chai Wah Wu, Dec 27 2014