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

Author

Clark Kimberling, Dec 07 2013

Keywords

• nonn
• easy

Comments

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

Examples

			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.
		

Links

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

Crossrefs

Cf. A233342.

Programs

Mathematica

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}]

Formula

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