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

1,2

Links

Table of n, a(n) for n=1..85.

Formula

Algorithm:

1. factorize n;

2. arrange prime factors by decreasing size;

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

Example

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.

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}]

Crossrefs

Cf. A084317-A084319, A085308, A085309.

Sequence in context: A136096 A284694 A072591 * A209190 A086286 A272565

Adjacent sequences: A085305 A085306 A085307 * A085309 A085310 A085311

Keyword

base,nonn

Author

Labos Elemer, Jun 27 2003

Status

approved