A385652 Maximum frequency of gpf(k) for 2 <= k <= n, where gpf(k) = A006530(k) is the greatest prime factor of k.

1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 12

Offset

2,3

Comments

The prime p is popular on the interval [2,n] if it is the greatest prime factor of a(n) numbers in that interval; see A385503.

Links

Pontus von Brömssen, Table of n, a(n) for n = 2..10000

Formula

a(n) = max_{k=2..n} A078899(k).

Example

     |     | cumulative frequencies for gpf's |
   n | gpf |    2  3  5  7 11 13 17 19 23     | a(n)
  ---+-----+----------------------------------+-----
   2 |   2 |    1  0  0  0  0  0  0  0  0     |  1
   3 |   3 |    1  1  0  0  0  0  0  0  0     |  1
   4 |   2 |    2  1  0  0  0  0  0  0  0     |  2
   5 |   5 |    2  1  1  0  0  0  0  0  0     |  2
   6 |   3 |    2  2  1  0  0  0  0  0  0     |  2
   7 |   7 |    2  2  1  1  0  0  0  0  0     |  2
   8 |   2 |    3  2  1  1  0  0  0  0  0     |  3
   9 |   3 |    3  3  1  1  0  0  0  0  0     |  3
  10 |   5 |    3  3  2  1  0  0  0  0  0     |  3
  11 |  11 |    3  3  2  1  1  0  0  0  0     |  3
  12 |   3 |    3  4  2  1  1  0  0  0  0     |  4
  13 |  13 |    3  4  2  1  1  1  0  0  0     |  4
  14 |   7 |    3  4  2  2  1  1  0  0  0     |  4
  15 |   5 |    3  4  3  2  1  1  0  0  0     |  4
  16 |   2 |    4  4  3  2  1  1  0  0  0     |  4
  17 |  17 |    4  4  3  2  1  1  1  0  0     |  4
  18 |   3 |    4  5  3  2  1  1  1  0  0     |  5
  19 |  19 |    4  5  3  2  1  1  1  1  0     |  5
  20 |   5 |    4  5  4  2  1  1  1  1  0     |  5
  21 |   7 |    4  5  4  3  1  1  1  1  0     |  5
  22 |  11 |    4  5  4  3  2  1  1  1  0     |  5
  23 |  23 |    4  5  4  3  2  1  1  1  1     |  5
  24 |   3 |    4  6  4  3  2  1  1  1  1     |  6

Prog

(Python)
from collections import Counter
from itertools import count
from sympy import factorint
def A385652_generator():
    c = Counter()
    M = 0
    for n in count(2):
        gpf = max(factorint(n))
        c[gpf] += 1
        if c[gpf] > M: M += 1
        yield M
(PARI) gpf(n) = if (n==1, 1, vecmax(factor(n)[, 1])); \\ A006530
a(n) = my(v=vector(n, k, gpf(k)), s=Set(v)); vecmax(apply(x->#x, vector(#s, i, select(x->(x==s[i]), v)))); \\ Michel Marcus, Jul 06 2025

Crossrefs

Cf. A006530, A078899, A385503, A385653, A385654.

Sequence in context: A056172 A285881 A091373 * A197637 A235492 A226762

Adjacent sequences: A385649 A385650 A385651 * A385653 A385654 A385655

Keyword

nonn

Author

Pontus von Brömssen, Jul 06 2025

Status

approved