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%I #11 May 20 2019 14:15:31
%S 1,2,4,3,16,8,256,5,9,32,65536,12,4294967296,512,64,6,
%T 18446744073709551616,18,340282366920938463463374607431768211456,48,
%U 1024,131072,115792089237316195423570985008687907853269984665640564039457584007913129639936,20,81,8589934592,25
%N Multiplicative with a(prime(k)^e) = A005117(e+1)^(2^(k-1)).
%C This sequence is a permutation of the natural numbers (with inverse A308328).
%C The property of being a bijection is easily deduced from the Fermi-Dirac representation of a number.
%C The first known fixed points are: 1, 2, 9, 12, 18, 3584, 32256.
%H OEIS Wiki, <a href="/wiki/%22Fermi-Dirac_representation%22_of_n">"Fermi-Dirac representation" of n</a>
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F a(2^e) = A005117(e+1) for any e >= 0.
%F a(prime(k)) = A001146(k-1) for any k > 0.
%F A000120(A267116(a(n))) = A001221(n).
%e a(3) = a(prime(2)) = A001146(1) = 2^(2^1) = 4.
%e a(2^5) = A005117(6) = 7.
%e a(96) = a(2^5 * 3) = a(2^5) * a(3) = 7 * 4 = 28.
%o (PARI) A005117(n) = for (k=1, oo, if (issquarefree(k), if (n--==0, return (k))))
%o a(n) = my (f=factor(n)); prod (i=1, #f~, A005117(1+f[i,2])^(2^(primepi(f[i,1])-1)))
%Y Cf. A000120, A001146, A001221, A005117, A267116, A308328 (inverse), A318363.
%K nonn,mult
%O 1,2
%A _Rémy Sigrist_, May 19 2019
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