A298882 a(1) = 1, and for any n > 1, if n is the k-th number with least prime factor p, then a(n) is the k-th number with greatest prime factor p.

1, 2, 3, 4, 5, 8, 7, 16, 6, 32, 11, 64, 13, 128, 9, 256, 17, 512, 19, 1024, 12, 2048, 23, 4096, 10, 8192, 18, 16384, 29, 32768, 31, 65536, 24, 131072, 15, 262144, 37, 524288, 27, 1048576, 41, 2097152, 43, 4194304, 36, 8388608, 47, 16777216, 14, 33554432, 48

Offset: 1

Author

Rémy Sigrist, Jan 28 2018

Keywords

• nonn

Comments

This sequence is a permutation of the natural numbers, with inverse A298268.

For any prime p and k > 0:

- if s_p(k) is the k-th p-smooth number and r_p(k) is the k-th p-rough number,

- then a(p * r_p(k)) = p * s_p(k),

- for example: a(11 * A008364(k)) = 11 * A051038(k).

Examples

			The first terms, alongside A020639(n), are:
  n     a(n)    lpf(n)
  --    ----    ------
   1       1      1
   2       2      2
   3       3      3
   4       4      2
   5       5      5
   6       8      2
   7       7      7
   8      16      2
   9       6      3
  10      32      2
  11      11     11
  12      64      2
  13      13     13
  14     128      2
  15       9      3
  16     256      2
  17      17     17
  18     512      2
  19      19     19
  20    1024      2
		

Links

• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A008364, A020639, A046022, A051038, A055396, A078898, A083140, A125624, A298268 (inverse).

Formula

a(1) = 1.

a(A083140(n, k)) = A125624(n, k) for any n > 0 and k > 0.

a(n) = A125624(A055396(n), A078898(n)) for any n > 1.

Empirically:

- a(n) = n iff n belongs to A046022,

- a(2 * k) = 2^k for any k > 0,

- a(p^2) = 2 * p for any prime p,

- a(p * q) = 3 * p for any pair of consecutive odd primes (p, q).