|
%I #24 Jun 25 2023 18:53:52
%S 1,4,8,12,25,30,32,48,63,70,75,80,165,176,189,192,289,306,315,320,385,
%T 392,507,520,575,598,621,644,841,858,957,968,1015,1044,1071,1088,1105,
%U 1122,1425,1444,1463,1470,1771,1782,1935,1978,2145,2156,2303,2350,2397
%N Least number m for which n applications of the mapping r(k) = k - (greatest prime divisor of k) map m to 0.
%C r(m) = 1 if and only if m = 1 or m is a prime. Conjecture: Every positive integer divides infinitely many terms of A233341.
%C 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
%C 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
%H Christian N. K. Anderson, <a href="/A233341/b233341.txt">Table of n, a(n) for n = 1..1000</a> (first 200 terms from Clark Kimberling)
%F For 228 <= n <= 1000, a(n) ~ 0.8526*n^2.023 to within 4% (empirical observation). - _Christian N. K. Anderson_, May 05 2023
%e 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.
%t 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}]
%Y Cf. A233342.
%K nonn,easy
%O 1,2
%A _Clark Kimberling_, Dec 07 2013
|
