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%I #27 Jul 27 2020 11:23:18
%S 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 27,28,11,30,79,32,14,57,35,36,37,69,22,40,101,42,17,44,45,87,47,48,
%U 49,50,26,52,53,54,55,56,34,93,139,60,61,111,63
%N 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 This is an involution on the natural numbers: applying it twice gives the identity permutation.
%H Chai Wah Wu, <a href="/A253047/b253047.txt">Table of n, a(n) for n = 1..10000</a>
%H 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 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%p f:= proc(t)
%p local r;
%p if t mod 2 = 0 and isprime(t/2) then 3*nextprime(t/2)
%p elif t mod 3 = 0 and isprime(t/3) then 2*prevprime(t/3)
%p elif isprime(t) then
%p r:= numtheory:-pi(t);
%p if isprime(r) then ithprime(2*r)
%p elif r mod 2 = 0 and isprime(r/2) then ithprime(r/2)
%p else t
%p fi
%p else t
%p fi
%p end proc:
%p seq(f(i),i=1..100); # _Robert Israel_, Dec 26 2014
%t 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 Array[f, 100] (* _Jean-François Alcover_, Jul 27 2020, after _Robert Israel_ *)
%o (Python)
%o from sympy import isprime, primepi, prevprime, nextprime, prime
%o def A253047(n):
%o ....if n <= 2:
%o ........return n
%o ....if n == 3:
%o ........return 7
%o ....q2, r2 = divmod(n,2)
%o ....if r2:
%o ........q3, r3 = divmod(n,3)
%o ........if r3:
%o ............if isprime(n):
%o ................m = primepi(n)
%o ................if isprime(m):
%o ....................return prime(2*m)
%o ................x, y = divmod(m,2)
%o ................if not y:
%o ....................if isprime(x):
%o ........................return prime(x)
%o ............return n
%o ........if isprime(q3):
%o ............return 2*prevprime(q3)
%o ........return n
%o ....if isprime(q2):
%o ........return 3*nextprime(q2)
%o ....return n # _Chai Wah Wu_, Dec 27 2014
%Y Cf. A064614, A253046, A253046.
%K nonn
%O 1,2
%A _N. J. A. Sloane_, Dec 26 2014
%E Definition supplied by _Robert Israel_, Dec 26 2014
%E Offset changed to 1 by _Chai Wah Wu_, Dec 27 2014
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