A233341 -id:A233341 - cp's OEIS frontend

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

Clark Kimberling, Dec 07 2013

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

			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

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

Cf. A233341, A000040, A100484.

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

A232724 Numbers k satisfying g(k - g(k)) > g(k) = greatest prime divisor of k.

Original entry on oeis.org

8, 16, 18, 24, 32, 36, 40, 48, 54, 60, 64, 72, 75, 81, 84, 90, 96, 98, 100, 108, 120, 126, 128, 135, 140, 144, 150, 154, 160, 162, 168, 180, 189, 192, 198, 200, 210, 216, 220, 224, 225, 234, 240, 243, 245, 250, 256, 260, 264, 270, 280, 288, 294, 297, 300

Offset: 1

Clark Kimberling, Dec 11 2013

Conjecture: for every positive integer d, there exist infinitely many n for which a(n + 1) - a(n) + d; for d = 1, the first 4 such n are 40, 67, 76, 79.

			g(18) = 3, g(18-3) = g(15) = 5, and 18 is the 3rd positive integer having the defining property, so a(3) = 18.

Clark Kimberling, Table of n, a(n) for n = 1..1502

Cf. A233341, A233342, A006530, A000040.

g[n_] := g[n] = FactorInteger[n][[-1, 1]]; t = {}; Do[If[g[n - g[n]] > g[n], AppendTo[t, n]], {n, 1, 500}]; t

cp's OEIS Frontend

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.

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