A249727 -id:A249727 - cp's OEIS frontend

A272616 Rectangular array, r(n,k), by antidiagonals: the interspersion associated with the fractal sequence A249727.

1, 2, 3, 4, 5, 8, 6, 7, 11, 15, 9, 10, 14, 19, 32, 12, 13, 18, 23, 37, 43, 16, 17, 22, 27, 42, 49, 68, 20, 21, 26, 31, 48, 55, 75, 83, 24, 25, 30, 36, 54, 61, 82, 91, 116, 28, 29, 35, 41, 60, 67, 90, 99, 125, 171, 33, 34, 40, 47, 66, 74, 98, 107, 134, 181

Offset: 1

Clark Kimberling, May 14 2016

r(n,k) is the position of the k-th occurrence of n in A249727. Every positive integer occurs exactly once, and each row is interspersed by each other row, except for initial terms.

			Northwest corner:
1     2     4     6     9     12    16    20
3     5     7     10    13    17    21    25
8     11    14    18    22    26    30    35
15    19    23    27    31    36    41    47
32    37    42    48    54    60    66    73
43    49    55    61    67    74    81    89

Cf. A249727, A061536 (= row 1).

t = Flatten[Table[Range[PrimePi[n]], {n, 2, 200}]];
r[n_, k_] := Flatten[Position[t, n]][[k]]
TableForm[Table[r[n, k], {n, 1, 12}, {k, 1, 12}]]  (* A272616 array*)
Table[r[n - k + 1, k], {n, 15}, {k, n, 1, -1}] // Flatten  (* A272616 sequence*)

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)).

Original entry on oeis.org

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)).

A249728 After a(1) = 1 each n appears A000720(n) times.

Original entry on oeis.org

1, 2, 3, 3, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19, 19, 19, 19, 20, 20, 20, 20, 20, 20, 20, 20

Offset: 1

Antti Karttunen, Nov 06 2014

nonn

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

Cf. A000720, A046992, A249727, A249809.

Join[{1},Table[Table[n,PrimePi[n]],{n,20}]//Flatten] (* Harvey P. Dale, Nov 24 2018 *)

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A272616 Rectangular array, r(n,k), by antidiagonals: the interspersion associated with the fractal sequence A249727.

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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)).

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A249728 After a(1) = 1 each n appears A000720(n) times.

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