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A233342 Rectangular array by antidiagonals: row n shows the numbers m for which n is the number of applications of the mapping r(k) = k - (greatest prime divisor or k) required to map m to 0.

%I A233342 #8 Dec 18 2013 23:36:19
%S A233342 1,2,4,3,6,8,5,10,9,12,7,14,15,18,25,11,22,16,20,27,30,13,26,21,24,35,
%T A233342 40,32,17,34,33,28,55,42,45,48,19,38,39,36,65,60,49,50,63,23,46,51,44,
%U A233342 85,66,77,56,99,70,29,58,57,52,95,78,81,84,105,108,75
%N A233342 Rectangular array by antidiagonals: row n shows the numbers m for which n is the number of applications of the mapping r(k) = k - (greatest prime divisor or k) required to map m to 0.
%C A233342 Every positive integer occurs exactly once in the array, so that the sequence is a permutation of the natural numbers.
%C A233342 Row 1:  A008578 (primes at the beginning of the 20th century)
%C A233342 Row 2:  A100484 (even semiprimes)
%C A233342 Col 1:  A233341
%H A233342 Clark Kimberling, <a href="/A233342/b233342.txt">Antidiagonals n = 1..60, flattened</a>
%e A233342 Northwest corner:
%e A233342 1 ... 2 ... 3 ... 5 ... 7 ... 11 .. 13
%e A233342 4 ... 6 ... 10 .. 14 .. 22 .. 26 .. 34
%e A233342 8 ... 9 ... 15 .. 16 .. 21 .. 33 .. 39
%e A233342 12 .. 18 .. 20 .. 24 .. 28 .. 36 .. 44
%e A233342 25 .. 27 .. 35 .. 55 .. 65 .. 85 .. 95
%e A233342 30 .. 40 .. 42 .. 60 .. 66 .. 78 .. 90
%t A233342 z = 40000; h[n_] := h[n] = n - FactorInteger[n][[-1, 1]]; t[n_] := Drop[FixedPointList[h, n], -2]; a = Table[Length[t[n]], {n, 1, z}] ; r[n_] := r[n] = Flatten[Position[a, n]]; w[n_, k_] := r[n][[k]]; TableForm[Table[w[n, k], {n, 1, 10}, {k, 1, 10}]]
%t A233342 u = Table[w[n - k + 1, k], {n, 14}, {k, n, 1, -1}] // Flatten
%Y A233342 Cf. A233341, A000040, A100484.
%K A233342 nonn,tabl,easy
%O A233342 1,2
%A A233342 _Clark Kimberling_, Dec 07 2013