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

1,3

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.

Links

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

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.

Example

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,
  ...

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A076259 A260533 A107359 * A342255 A112377 A354521

Adjacent sequences: A307638 A307639 A307640 * A307642 A307643 A307644

Keyword

nonn,tabl

Author

I. V. Serov, Apr 19 2019

Status

approved