A369609 Irregular triangle read by rows where row n lists k <= n such that A007947(k) = A007947(n).

1, 2, 3, 2, 4, 5, 6, 7, 2, 4, 8, 3, 9, 10, 11, 6, 12, 13, 14, 15, 2, 4, 8, 16, 17, 6, 12, 18, 19, 10, 20, 21, 22, 23, 6, 12, 18, 24, 5, 25, 26, 3, 9, 27, 14, 28, 29, 30, 31, 2, 4, 8, 16, 32, 33, 34, 35, 6, 12, 18, 24, 36, 37, 38, 39, 10, 20, 40, 41, 42, 43, 22, 44

Offset: 1

Author

Michael De Vlieger, May 09 2024

Keywords

• nonn
• tabf

Comments

Differs from A284318 after 27 terms.

Let rad(x) = A007947(x).

Let T(n,k) be the k-th term of row n in this sequence.

Define S(n,k) to be the k-th term in row n of A162306.

T(n,k) = rad(n) * S(n,k), k <= A008479(n).

The number n appears as the last term in row n.

Examples

			First rows of the triangle:
  1;
  2;
  3;
  2, 4;
  5;
  6;
  7;
  2, 4, 8;
  3, 9;
  10;
  11;
  6, 12;
  13;
  14;
  15;
  2, 4, 8, 16;
  17;
  6, 12, 18;
  etc.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..12946 (rows n = 1..5000, flattened)
• Michael De Vlieger, Fast Mathematica programs that construct A008479, A010846, A162306, and A369609

Crossrefs

Cf. A007947, A008479, A162306, A284318.

Programs

Mathematica

f[x_] := f[x] = Times @@ FactorInteger[x][[All, 1]]; Flatten@ Table[r = f[n]; Select[Range[n], f[#] == r &], {n, 44}]

PARI

rad(n) = factorback(factorint(n)[, 1]); \\ A007947
row(n) = my(r=rad(n)); select(x->(rad(x) == r), [1..n]); \\ Michel Marcus, May 11 2024

Formula

Row n of this sequence contains row n of A284318.

Length of row n is A008479(n).

For squarefree n, row n = {n}.

For prime power n = p^m, row n = { p^j : j = 1..m }.