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

1,5

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

Cf. A133457.

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

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

Cf. A082786 (same as regular triangle).

For other triangle versions see A060175, A143078.

Cf. A054841, rows reversed and concatenated into a decimal number.

Sequence in context: A064559 A340998 A336562 * A065716 A079409 A369461

Adjacent sequences: A067252 A067253 A067254 * A067256 A067257 A067258

Keyword

easy,nonn,tabf

Author

Jeppe Stig Nielsen, Feb 20 2002

Status

approved