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A320252 Numbers with prime factorization Product_{k=1..w} prime(i_k) ^ e_k (where w = A001221(n) and prime(i) denotes the i-th prime number) such that i_k <> e_k for k = 1..w and { i_1, ..., i_w } = { e_1, ..., e_w }.

%I A320252 #11 Jan 03 2021 15:11:25
%S A320252 1,12,40,112,352,540,600,675,832,2176,2268,2352,3969,4864,10692,11616,
%T A320252 11776,27440,29403,29696,32448,35000,37908,63488,75600,105840,110976,
%U A320252 113400,123201,148716,151552,158760,212960,214375,237600,275000,277248,335872,411600
%N A320252 Numbers with prime factorization Product_{k=1..w} prime(i_k) ^ e_k (where w = A001221(n) and prime(i) denotes the i-th prime number) such that i_k <> e_k for k = 1..w and { i_1, ..., i_w } = { e_1, ..., e_w }.
%C A320252 This sequence is a subsequence of A109297.
%C A320252 For any i > 0 and j > 0 such that a(i) and a(j) are coprime, a(i) * a(j) belongs to this sequence.
%C A320252 For any i > 0, A048767(a(i)) belongs to this sequence.
%C A320252 Let S be the set of permutations of the natural numbers with finitely many non-fixed points:
%C A320252 - we can build a bijection f from S to this sequence as follows: for any s in S, f(s) = Product_{s(i) <> i} prime(i) ^ s(i),
%C A320252 - for any s in S with inverse z, f(z) = A048767(f(s)).
%F A320252 A001221(a(n)) = A071625(a(n)).
%e A320252 The first terms, alongside the corresponding permutations, are:
%e A320252   n   a(n)    s
%e A320252   --  ------  ----------
%e A320252    1       1  ()
%e A320252    2      12  (1 2)
%e A320252    3      40  (1 3)
%e A320252    4     112  (1 4)
%e A320252    5     352  (1 5)
%e A320252    6     540  (1 2 3)
%e A320252    7     600  (1 3 2)
%e A320252    8     675  (2 3)
%e A320252    9     832  (1 6)
%e A320252   10    2176  (1 7)
%e A320252   11    2268  (1 2 4)
%e A320252   12    2352  (1 4 2)
%e A320252   13    3969  (2 4)
%e A320252   14    4864  (1 8)
%e A320252   15   10692  (1 2 5)
%e A320252   16   11616  (1 5 2)
%e A320252   17   11776  (1 9)
%e A320252   18   27440  (1 4 3)
%e A320252   19   29403  (2 5)
%e A320252   20   29696  (1 10)
%e A320252   21   32448  (1 6 2)
%e A320252   22   35000  (1 3 4)
%e A320252   23   37908  (1 2 6)
%e A320252   24   63488  (1 11)
%e A320252   25   75600  (1 4)(2 3)
%o A320252 (PARI) is(n) = my (f=factor(n), i=apply(primepi, f[,1]~), e=f[,2]~); #select(k -> i[k]==e[k], [1..#f~])==0 && Set(i) == Set(e)
%Y A320252 Cf. A001221, A048767, A071625, A109297.
%K A320252 nonn
%O A320252 1,2
%A A320252 _Rémy Sigrist_, Oct 08 2018