A085308 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.
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, 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, 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
Offset: 1
Examples
n=even: remains even: m = 100 = 2*2*5*5 -> {2,5} -> {5,2} -> 52 = a(100);
n = 2^i*3^j: a(n)=2 since iteration list is {n,32,2}; these
are the known convergent even cases of initial value.
n=143: a(143) = 44864859110711 because the iteration list is
{143, 1311, 23193, 8593, 66113, 388917, 547793, 2273241, 55311373, 989474313, 8914183373, 84859143973, 528059391607, 44864859110711};
a(n) = 0 for n = 213, 323, 639, 713 ending in {713, 3123, 3473, 15123}; terminal orbit of length = 4.
All possible cases occur: fixed point, divergence, terminal cycle.
Programs
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Mathematica
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}]
Formula
Algorithm:
1. factorize n;
2. arrange prime factors by decreasing size;
3. concatenate prime factors and interpret the result as a decimal number.