A233342 -id:A233342 - cp's OEIS frontend

A233341 Least number m for which n applications of the mapping r(k) = k - (greatest prime divisor of k) map m to 0.

1, 4, 8, 12, 25, 30, 32, 48, 63, 70, 75, 80, 165, 176, 189, 192, 289, 306, 315, 320, 385, 392, 507, 520, 575, 598, 621, 644, 841, 858, 957, 968, 1015, 1044, 1071, 1088, 1105, 1122, 1425, 1444, 1463, 1470, 1771, 1782, 1935, 1978, 2145, 2156, 2303, 2350, 2397

Offset: 1

Clark Kimberling, Dec 07 2013

r(m) = 1 if and only if m = 1 or m is a prime. Conjecture: Every positive integer divides infinitely many terms of this sequence.
Sequence is empirically observed to be strictly increasing for n <= 1000, in contrast to similar map in A050710. - Christian N. K. Anderson, May 05 2023
Observe that for ~4/7 of the first thousand terms, r(a(n)) = a(n-1); e.g., a(12)=80, r(80)=75=a(11) -> 70=a(10) -> 63=a(9). However, the other ~3/7 take a different route to zero; e.g., a(9)=63 decreases by 7 at all 9 steps. Contrast A048133, where every term's iteration ends when r(k)=5. - Christian N. K. Anderson, May 05 2023

			r(8) = 8 - 2 = 6; r(6) = 6 - 3 = 3; r(3) = 3 - 3 = 0. Thus 3 applications of r map 8 to 0, whereas 1 or 2 applications suffice for n < 8. Therefore, a(3) = 8.

Christian N. K. Anderson, Table of n, a(n) for n = 1..1000 (first 200 terms from Clark Kimberling)

Cf. A233342.

z = 10000; h[n_] := h[n] = n - FactorInteger[n][[-1, 1]]; t[n_] := Drop[FixedPointList[h, n], -2]; Table[t[n], {n, 1, z}]; a = Table[Length[t[n]], {n, 1, z}]; f[n_] := First[Flatten[Position[a, n]]]; Table[f[n], {n, 1, 80}]

For 228 <= n <= 1000, a(n) ~ 0.8526*n^2.023 to within 4% (empirical observation). - Christian N. K. Anderson, May 05 2023

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

A233341 Least number m for which n applications of the mapping r(k) = k - (greatest prime divisor of k) map m to 0.

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