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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.

%I A253047 #29 May 22 2025 10:21:41
%S A253047 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,
%T A253047 27,28,11,30,79,32,14,57,35,36,37,69,22,40,101,42,17,44,45,87,47,48,
%U A253047 49,50,26,52,53,54,55,56,34,93,139,60,61,111,63
%N 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.
%C A253047 This is an involution on the natural numbers: applying it twice gives the identity permutation.
%H A253047 Chai Wah Wu, <a href="/A253047/b253047.txt">Table of n, a(n) for n = 1..10000</a>
%H A253047 A. B. Frizell, <a href="http://dx.doi.org/10.1090/S0002-9904-1915-02686-8">Certain non-enumerable sets of infinite permutations</a>. Bull. Amer. Math. Soc. 21 (1915), no. 10, 495-499.
%H A253047 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%p A253047 f:= proc(t)
%p A253047 local r;
%p A253047   if t mod 2 = 0 and isprime(t/2) then  3*nextprime(t/2)
%p A253047   elif t mod 3 = 0 and isprime(t/3) then 2*prevprime(t/3)
%p A253047   elif isprime(t) then
%p A253047     r:= numtheory:-pi(t);
%p A253047     if isprime(r) then ithprime(2*r)
%p A253047     elif r mod 2 = 0 and isprime(r/2) then ithprime(r/2)
%p A253047     else t
%p A253047     fi
%p A253047   else t
%p A253047   fi
%p A253047 end proc:
%p A253047 seq(f(i),i=1..100); # _Robert Israel_, Dec 26 2014
%t A253047 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]];
%t A253047 Array[f, 100] (* _Jean-François Alcover_, Jul 27 2020, after _Robert Israel_ *)
%o A253047 (Python)
%o A253047 from sympy import isprime, primepi, prevprime, nextprime, prime
%o A253047 def A253047(n):
%o A253047     if n <= 2:
%o A253047         return n
%o A253047     if n == 3:
%o A253047         return 7
%o A253047     q2, r2 = divmod(n,2)
%o A253047     if r2:
%o A253047         q3, r3 = divmod(n,3)
%o A253047         if r3:
%o A253047             if isprime(n):
%o A253047                 m = primepi(n)
%o A253047                 if isprime(m):
%o A253047                     return prime(2*m)
%o A253047                 x, y = divmod(m,2)
%o A253047                 if not y:
%o A253047                     if isprime(x):
%o A253047                         return prime(x)
%o A253047             return n
%o A253047         if isprime(q3):
%o A253047             return 2*prevprime(q3)
%o A253047         return n
%o A253047     if isprime(q2):
%o A253047         return 3*nextprime(q2)
%o A253047     return n # _Chai Wah Wu_, Dec 27 2014
%Y A253047 Cf. A064614, A253046, A253046.
%K A253047 nonn
%O A253047 1,2
%A A253047 _N. J. A. Sloane_, Dec 26 2014
%E A253047 Definition supplied by _Robert Israel_, Dec 26 2014
%E A253047 Offset changed to 1 by _Chai Wah Wu_, Dec 27 2014