A377485 Irregular triangle where row n lists powers p^k of primes p | n such that p^k <= n and k > 0.

1, 2, 3, 2, 4, 5, 2, 3, 4, 7, 2, 4, 8, 3, 9, 2, 4, 5, 8, 11, 2, 3, 4, 8, 9, 13, 2, 4, 7, 8, 3, 5, 9, 2, 4, 8, 16, 17, 2, 3, 4, 8, 9, 16, 19, 2, 4, 5, 8, 16, 3, 7, 9, 2, 4, 8, 11, 16, 23, 2, 3, 4, 8, 9, 16, 5, 25, 2, 4, 8, 13, 16, 3, 9, 27, 2, 4, 7, 8, 16, 29

Offset: 1

Author

Michael De Vlieger, Oct 29 2024

Keywords

• nonn
• easy
• tabf

Comments

Row 1 is {1} by convention, since 1 is the empty product.

Examples

			Table of the first 12 rows:
   n   row n
  -------------------
   1:  1;
   2:  2;
   3:  3:
   4:  2, 4;
   5:  5;
   6:  2, 3, 4;
   7:  7;
   8:  2, 4, 8;
   9:  3, 9;
  10:  2, 4, 5, 8;
  11: 11;
  12:  2, 3, 4, 8, 9;
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..12747 (rows n = 1..1500, flattened.)

Crossrefs

Cf. A162306, A246655.

Programs

Mathematica

{{1}}~Join~Table[Union[Join @@ Map[#^Range[Floor@ Log[#, n]] &, FactorInteger[n][[All, 1]] ] ], {n, 2, 30}]

Formula

Row n is { p^k : p | n, k = 1..floor(log n/log p) }, i.e., intersection of A246655 and row n of A162306.

Row p = {p} for prime p.

Row p^k = { p^j : j = 1..k } for prime p and k > 0.

A361373(n) = length of row n for n > 1.