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

Offset: 1

Author

Jeppe Stig Nielsen, Feb 20 2002

Keywords

• easy
• nonn
• tabf

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

Examples

			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.
		

Links

• Reinhard Zumkeller, Rows n = 1..250 of triangle, flattened
• Jeppe Stig Nielsen, See this explanation.

Crossrefs

Cf. A133457.

Cf. A001222 (row sums), A061395 (lengths of rows n >= 2).

Cf. A007814 (left edge), A071178 (right edge).

Other versions: A054841 (rows reversed and concatenated into a decimal number), A060175 (square array), A082786 (regular triangle), A124010 (with 0's removed, excepting row 1), A143078 (another irregular triangle).

Programs

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

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 *)