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 }.

1, 12, 40, 112, 352, 540, 600, 675, 832, 2176, 2268, 2352, 3969, 4864, 10692, 11616, 11776, 27440, 29403, 29696, 32448, 35000, 37908, 63488, 75600, 105840, 110976, 113400, 123201, 148716, 151552, 158760, 212960, 214375, 237600, 275000, 277248, 335872, 411600

Offset: 1

Author

Rémy Sigrist, Oct 08 2018

Keywords

• nonn

Comments

This sequence is a subsequence of A109297.

For any i > 0 and j > 0 such that a(i) and a(j) are coprime, a(i) * a(j) belongs to this sequence.

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

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

- 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),

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

Examples

			The first terms, alongside the corresponding permutations, are:
  n   a(n)    s
  --  ------  ----------
   1       1  ()
   2      12  (1 2)
   3      40  (1 3)
   4     112  (1 4)
   5     352  (1 5)
   6     540  (1 2 3)
   7     600  (1 3 2)
   8     675  (2 3)
   9     832  (1 6)
  10    2176  (1 7)
  11    2268  (1 2 4)
  12    2352  (1 4 2)
  13    3969  (2 4)
  14    4864  (1 8)
  15   10692  (1 2 5)
  16   11616  (1 5 2)
  17   11776  (1 9)
  18   27440  (1 4 3)
  19   29403  (2 5)
  20   29696  (1 10)
  21   32448  (1 6 2)
  22   35000  (1 3 4)
  23   37908  (1 2 6)
  24   63488  (1 11)
  25   75600  (1 4)(2 3)
		

Crossrefs

Cf. A001221, A048767, A071625, A109297.

Programs

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)

Formula

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