A067255 - OEIS

A067255

Irregular triangle read by rows: row n gives exponents in prime factorization of n.

0, 1, 0, 1, 2, 0, 0, 1, 1, 1, 0, 0, 0, 1, 3, 0, 2, 1, 0, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 4, 0, 0, 0, 0, 0, 0, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 1, 0, 0, 2, 1, 0, 0, 0, 0, 1, 0, 3, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

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OFFSET

1,5

COMMENTS

Row lengths are given by A061395(n), n >= 2: [1, 2, 1, 3, 2, 4, 1, 2, ... ].

This sequence contains every finite sequence of nonnegative integers. - Franklin T. Adams-Watters, Jun 22 2005

LINKS

Reinhard Zumkeller, Rows n = 1..250 of triangle, flattened

Jeppe Stig Nielsen, See this explanation.

EXAMPLE

1 = 2^0

2 = 2^1

3 = 2^0 3^1

4 = 2^2

5 = 2^0 3^0 5^1

6 = 2^1 3^1

... and reading the exponents gives the sequence.

Since for example 99=2^0*3^2*5^0*7^0*11^1, we use this symbol for ninety-nine: 99: {0,2,0,0,1}. Concatenating all the symbols for 1,2,3,4,5,6,..., we get the sequence.

MATHEMATICA

f[n_] := If[n == 1, {0}, Function[f, ReplacePart[Table[0, {PrimePi[f[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, f]]@ FactorInteger@ n]; Array[f, 29] // Flatten (* Michael De Vlieger, Mar 08 2019 *)

PROG

(Haskell)

a067255 n k = a067255_tabf !! (n-1) !! (k-1)

a067255_row 1 = [0]

a067255_row n = f n a000040_list where

f 1 _ = []

f u (p:ps) = g u 0 where

g v e = if m == 0 then g v' (e + 1) else e : f v ps

where (v', m) = divMod v p

a067255_tabf = map a067255_row [1..]

-- Reinhard Zumkeller, Jun 11 2013

CROSSREFS