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.

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, 40, 32, 17, 34, 33, 28, 55, 42, 45, 48, 19, 38, 39, 36, 65, 60, 49, 50, 63, 23, 46, 51, 44, 85, 66, 77, 56, 99, 70, 29, 58, 57, 52, 95, 78, 81, 84, 105, 108, 75

Offset: 1

Author

Clark Kimberling, Dec 07 2013

Keywords

• nonn
• tabl
• easy

Comments

Every positive integer occurs exactly once in the array, so that the sequence is a permutation of the natural numbers.

Row 1: A008578 (primes at the beginning of the 20th century)

Row 2: A100484 (even semiprimes)

Col 1: A233341

Examples

			Northwest corner:
1 ... 2 ... 3 ... 5 ... 7 ... 11 .. 13
4 ... 6 ... 10 .. 14 .. 22 .. 26 .. 34
8 ... 9 ... 15 .. 16 .. 21 .. 33 .. 39
12 .. 18 .. 20 .. 24 .. 28 .. 36 .. 44
25 .. 27 .. 35 .. 55 .. 65 .. 85 .. 95
30 .. 40 .. 42 .. 60 .. 66 .. 78 .. 90
		

Links

• Clark Kimberling, Antidiagonals n = 1..60, flattened

Crossrefs

Cf. A233341, A000040, A100484.

Programs

Mathematica

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}]]
u = Table[w[n - k + 1, k], {n, 14}, {k, n, 1, -1}] // Flatten