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%I #39 Aug 27 2023 08:43:16
%S 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,
%T 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,
%U 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
%N Irregular triangle read by rows: row n gives exponents in prime factorization of n.
%C Row lengths are given by A061395(n), n >= 2: [1, 2, 1, 3, 2, 4, 1, 2, ... ].
%C This sequence contains every finite sequence of nonnegative integers. - _Franklin T. Adams-Watters_, Jun 22 2005
%H Reinhard Zumkeller, <a href="/A067255/b067255.txt">Rows n = 1..250 of triangle, flattened</a>
%H Jeppe Stig Nielsen, <a href="http://jeppesn.dk/prim-eksp.text">See this explanation.</a>
%e 1 = 2^0
%e 2 = 2^1
%e 3 = 2^0 3^1
%e 4 = 2^2
%e 5 = 2^0 3^0 5^1
%e 6 = 2^1 3^1
%e ... and reading the exponents gives the sequence.
%e 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.
%t 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 *)
%o (Haskell)
%o a067255 n k = a067255_tabf !! (n-1) !! (k-1)
%o a067255_row 1 = [0]
%o a067255_row n = f n a000040_list where
%o f 1 _ = []
%o f u (p:ps) = g u 0 where
%o g v e = if m == 0 then g v' (e + 1) else e : f v ps
%o where (v',m) = divMod v p
%o a067255_tabf = map a067255_row [1..]
%o -- _Reinhard Zumkeller_, Jun 11 2013
%Y Cf. A133457.
%Y Cf. A001222 (row sums), A061395 (lengths of rows n >= 2).
%Y Cf. A007814 (left edge), A071178 (right edge).
%Y Cf. A082786 (same as regular triangle).
%Y For other triangle versions see A060175, A143078.
%Y Cf. A054841, rows reversed and concatenated into a decimal number.
%K easy,nonn,tabf
%O 1,5
%A _Jeppe Stig Nielsen_, Feb 20 2002
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