A298268 a(1) = 1, and for any n > 1, if n is the k-th number with greatest prime factor p, then a(n) is the k-th number with least prime factor p.

1, 2, 3, 4, 5, 9, 7, 6, 15, 25, 11, 21, 13, 49, 35, 8, 17, 27, 19, 55, 77, 121, 23, 33, 65, 169, 39, 91, 29, 85, 31, 10, 143, 289, 119, 45, 37, 361, 221, 95, 41, 133, 43, 187, 115, 529, 47, 51, 161, 125, 323, 247, 53, 57, 209, 203, 437, 841, 59, 145, 61, 961

Offset: 1

Author

Rémy Sigrist, Jan 27 2018

Keywords

• nonn

Comments

This sequence is a permutation of the natural numbers, with inverse A298882.

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 * s_p(k)) = p * r_p(k),

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

Examples

			The first terms, alongside A006530(n), are:
  n     a(n)   gpf(n)
  --    ----   ------
   1      1      1
   2      2      2
   3      3      3
   4      4      2
   5      5      5
   6      9      3
   7      7      7
   8      6      2
   9     15      3
  10     25      5
  11     11     11
  12     21      3
  13     13     13
  14     49      7
  15     35      5
  16      8      2
  17     17     17
  18     27      3
  19     19     19
  20     55      5
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, Colored logarithmic scatterplot of the first 100000 terms (with prime and semiprime values highlighted)
• Rémy Sigrist, Colored logarithmic scatterplot of the first 100000 terms (where the color is function of A006530(n))
• Rémy Sigrist, Colored logarithmic scatterplot of the first 100000 terms (where the color is function of A176506(k) when a(n) is the k-th semiprime)
• Rémy Sigrist, PARI program for A298268
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A006530, A008364, A046022, A051038, A061395, A078899, A083140, A125624, A151800, A176506, A298268, A298882 (inverse).

Programs

PARI

See Links section.

Formula

a(1) = 1.

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

a(n) = A083140(A061395(n), A078899(n)) for any n > 1.

Empirically:

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

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

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

- a(3 * p) = p * A151800(p) for any odd prime p.