A085308 - OEIS

%I #11 Sep 04 2017 23:25:58

%S 0,2,3,2,5,2,7,2,3,1,11,2,13,2,53,2,17,2,19,1,73,2,23,2,5,1,3,2,29,1,

%T 31,2,113,2,53,2,37,2,197,1,41,1,43,2,53,1,47,2,7,1,173,1,53,2,41113,

%U 2,193,1,59,1,61,1,73,2,53,1,67,1,233,1,2,73,1,53,1,197,1,79,1,3,1,83,1,53,1

%N Iterate function described in A085308 (= reverse concatenation of prime factors); a(n) is either 1# the fixed point[=prime] if it exists at all: 2# a(2k)=1 labels that no convergence with most even initial values, in contrary mostly rapid divergence is the case; 3# a(n)=0 if n=1 or if the iteration results in nontrivial attractor with cycle length larger than one.

%F Algorithm:

%F 1. factorize n;

%F 2. arrange prime factors by decreasing size;

%F 3. concatenate prime factors and interpret the result as a decimal number.

%e n=even: remains even: m = 100 = 2*2*5*5 -> {2,5} -> {5,2} -> 52 = a(100);

%e n = 2^i*3^j: a(n)=2 since iteration list is {n,32,2}; these

%e are the known convergent even cases of initial value.

%e n=143: a(143) = 44864859110711 because the iteration list is

%e {143, 1311, 23193, 8593, 66113, 388917, 547793, 2273241, 55311373, 989474313, 8914183373, 84859143973, 528059391607, 44864859110711};

%e a(n) = 0 for n = 213, 323, 639, 713 ending in {713, 3123, 3473, 15123}; terminal orbit of length = 4.

%e All possible cases occur: fixed point, divergence, terminal cycle.

%t ffi[x_] := Flatten[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] lf[x_] := Length[FactorInteger[x]] nd[x_, y_] := 10*x+y tn[x_] := Fold[nd, 0, x] rec[x_] := Fold[nd, 0, Flatten[IntegerDigits[Reverse[ba[x]]], 1]] Table[rec[w], {w, 1, 128}]

%Y Cf. A084317-A084319, A085308, A085309.

%K base,nonn

%O 1,2

%A _Labos Elemer_, Jun 27 2003