A249728 -id:A249728 - cp's OEIS frontend

A249809 Irregular table read by rows: T(n, k) is the number of times prime p_k has occurred as the smallest prime factor of numbers 1 .. n. (T(1,1) = 0, and for each n > 1, k = 1 .. A000720(n)).

0, 1, 1, 1, 2, 1, 2, 1, 1, 3, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 4, 2, 1, 1, 5, 2, 1, 1, 5, 2, 1, 1, 1, 6, 2, 1, 1, 1, 6, 2, 1, 1, 1, 1, 7, 2, 1, 1, 1, 1, 7, 3, 1, 1, 1, 1, 8, 3, 1, 1, 1, 1, 8, 3, 1, 1, 1, 1, 1, 9, 3, 1, 1, 1, 1, 1, 9, 3, 1, 1, 1, 1, 1, 1, 10, 3, 1, 1, 1, 1, 1, 1, 10, 4, 1, 1, 1, 1, 1, 1, 11, 4, 1, 1, 1, 1, 1, 1, 11, 4, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Nov 06 2014

After the first row {0}, consists of rows of triangular table A249808 with trailing zeros removed.

			Table begins:
       k=1  2  3  4  5  6  7
  n=1:   0;
  n=2:   1;
  n=3:   1, 1;
  n=4:   2, 1;
  n=5:   2, 1, 1;
  n=6:   3, 1, 1;
  n=7:   3, 1, 1, 1;
  n=8:   4, 1, 1, 1;
  n=9:   4, 2, 1, 1;
  n=10:  5, 2, 1, 1;
  n=11:  5, 2, 1, 1, 1;
  n=12:  6, 2, 1, 1, 1;
  n=13:  6, 2, 1, 1, 1, 1;
  n=14:  7, 2, 1, 1, 1, 1;
  n=15:  7, 3, 1, 1, 1, 1;
  n=16:  8, 3, 1, 1, 1, 1;
  n=17:  8, 3, 1, 1, 1, 1, 1;
  ...

Antti Karttunen, Table of n, a(n) for n = 1..10016; the first 294 rows of the table, flattened
Orges Leka, Fractal of prime factorization of integers, lexicographically sorted.
Orges Leka, How to define a fractal from the lexicographic sorting on the prime factorization of natural numbers?

A004526 gives the left edge, A001477 the row sums.

Cf. A000040, A000720, A020639, A249808, A249727, A249728, A055396, A078898.

(define (A249809 n) (A249808bi (A249728 n) (A249727 n))) ;; Code for A249808bi given in A249808.

a(n) = A249808(A249728(n), A249727(n)).

For n > 1, A078898(n) = T(n, A055396(n)).

A249727 Start with a(1) = 1; then numbers 1 .. primepi(2), followed by numbers 1 .. primepi(3), and then numbers 1 .. primepi(4), ..., etc, where A000720 gives primepi.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 3, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9

Offset: 1

Antti Karttunen, Nov 06 2014

nonn

Can be used to construct the irregular table A249809.

This is a fractal sequence; i.e., the removal of the first occurrence of each term in A249727 leaves A249727, so that the sequence contains itself infinitely many times. The corresponding interspersion is A272616. - Clark Kimberling, May 12 2016

Antti Karttunen, Table of n, a(n) for n = 1..10016

Cf. A000720, A046992, A249728, A249809, A272616.

Flatten[Table[Range[PrimePi[n]], {n, 2, 100}]]
(* Clark Kimberling, May 14 2016 *)

(define (A249727 n) (if (= 1 n) 1 (- n (+ 1 (A046992 (- (A249728 n) 1))))))

cp's OEIS Frontend

A249809 Irregular table read by rows: T(n, k) is the number of times prime p_k has occurred as the smallest prime factor of numbers 1 .. n. (T(1,1) = 0, and for each n > 1, k = 1 .. A000720(n)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Scheme

Formula