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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 }.
0
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
1,2
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)).
FORMULA
A001221(a(n)) = A071625(a(n)).
EXAMPLE
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)
PROG
(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)
CROSSREFS
KEYWORD
nonn
AUTHOR
Rémy Sigrist, Oct 08 2018
STATUS
approved