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A307641 Triangle T(i,j=1..i) read by rows which contain the naturally ordered prime-or-one factorization of the row number i. 5
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 (list; table; graph; refs; listen; history; text; internal format)
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 * A112377 A277760 A127704

Adjacent sequences:  A307638 A307639 A307640 * A307642 A307643 A307644

KEYWORD

nonn,tabl

AUTHOR

I. V. Serov, Apr 19 2019

STATUS

approved

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Last modified August 9 18:32 EDT 2020. Contains 336326 sequences. (Running on oeis4.)