A320252 - OEIS

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 #11 Jan 03 2021 15:11:25

%S 1,12,40,112,352,540,600,675,832,2176,2268,2352,3969,4864,10692,11616,

%T 11776,27440,29403,29696,32448,35000,37908,63488,75600,105840,110976,

%U 113400,123201,148716,151552,158760,212960,214375,237600,275000,277248,335872,411600

%N 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 This sequence is a subsequence of A109297.

%C 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 For any i > 0, A048767(a(i)) belongs to this sequence.

%C Let S be the set of permutations of the natural numbers with finitely many non-fixed points:

%C - 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 - for any s in S with inverse z, f(z) = A048767(f(s)).

%F A001221(a(n)) = A071625(a(n)).

%e The first terms, alongside the corresponding permutations, are:

%e n a(n) s

%e -- ------ ----------

%e 1 1 ()

%e 2 12 (1 2)

%e 3 40 (1 3)

%e 4 112 (1 4)

%e 5 352 (1 5)

%e 6 540 (1 2 3)

%e 7 600 (1 3 2)

%e 8 675 (2 3)

%e 9 832 (1 6)

%e 10 2176 (1 7)

%e 11 2268 (1 2 4)

%e 12 2352 (1 4 2)

%e 13 3969 (2 4)

%e 14 4864 (1 8)

%e 15 10692 (1 2 5)

%e 16 11616 (1 5 2)

%e 17 11776 (1 9)

%e 18 27440 (1 4 3)

%e 19 29403 (2 5)

%e 20 29696 (1 10)

%e 21 32448 (1 6 2)

%e 22 35000 (1 3 4)

%e 23 37908 (1 2 6)

%e 24 63488 (1 11)

%e 25 75600 (1 4)(2 3)

%o (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 Cf. A001221, A048767, A071625, A109297.

%K nonn

%O 1,2

%A _Rémy Sigrist_, Oct 08 2018