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

1,2

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).

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

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.

Example

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

Prog

(PARI) See Links section.

Crossrefs

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

Sequence in context: A346098 A245821 A075161 * A329044 A222251 A347759

Adjacent sequences: A298265 A298266 A298267 * A298269 A298270 A298271

Keyword

nonn

Author

Rémy Sigrist, Jan 27 2018

Status

approved