A307641 Triangle T(i,j=1..i) read by rows which contain the naturally ordered prime-or-one factorization of the row number i.

1, 1, 2, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 1, 5, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 2, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13

Offset: 1

Author

I. V. Serov, Apr 19 2019

Keywords

• nonn
• tabl

Comments

i=Product_{j=1..i} T(i,j). This is an adjusted formulation of the fundamental theorem of arithmetic with the fixed order of the prime-or-one factors, as well as with the regular length i of the factorization of i.

Remove all 1's except for n = 1 to get irregular triangle A307746.

A307723 is a quasi-logarithmic binary encoding of this triangle.

Examples

			Triangle begins:
  1,
  1, 2,
  1, 1, 3,
  1, 2, 1, 2,
  1, 1, 1, 1, 5,
  1, 2, 3, 1, 1, 1,
  1, 1, 1, 1, 1, 1, 7,
  1, 2, 1, 2, 1, 1, 1, 2,
  1, 1, 3, 1, 1, 1, 1, 1, 3,
  1, 2, 1, 1, 5, 1, 1, 1, 1, 1,
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1,11,
  1, 2, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1,
  ...
		

Links

• I. V. Serov, Rows n=1..131 of triangle, flattened

Crossrefs

Cf. A027746, A027748, A014963, A100995, A307662, A307723, A307742, A307743, A307746.

Programs

Mathematica

Table[Map[Which[PrimeNu@ # > 1, 1, And[PrimeQ@ #, Mod[n, #] == 0], #, Mod[n, #] == 0, FactorInteger[#][[1, 1]], True, 1] &, Range@ n], {n, 13}] // Flatten (* Michael De Vlieger, Apr 23 2019 *)

PARI

w(n) = my(t=isprimepower(n)); if (t, t, 0);
row(n) = vector(n, k, mnk = if ((n % k) == 0, k, 1); if (t=w(k), sqrtnint(mnk, t), 1)); \\ Michel Marcus, Apr 21 2019

Formula

T(i,j) = A307662(i,j)^w(j), where w(j)=0 if A100995(j)=0; otherwise w(j)=1/A100995(j), for 1 <= j <= n.