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

1,2

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).

Links

Table of n, a(n) for n=1..51.

Index entries for sequences that are permutations of the natural numbers

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).

Example

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

Crossrefs

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

Sequence in context: A075162 A056240 A069968 * A086931 A243405 A164339

Adjacent sequences: A298879 A298880 A298881 * A298883 A298884 A298885

Keyword

nonn

Author

Rémy Sigrist, Jan 28 2018

Status

approved