A084318 Iterate function described in A084317 if started at initial value n until reaching a fixed point.

0, 2, 3, 2, 5, 23, 7, 2, 3, 5, 11, 23, 13, 3, 1129, 2, 17, 23, 19, 5, 37, 211, 23, 23, 5, 3251, 3, 3, 29, 547, 31, 2, 311, 31397, 1129, 23, 37, 373, 313, 5, 41, 379, 43, 211, 1129, 223, 47, 23, 7, 5, 317, 3251, 53, 23, 773, 3, 1129, 229, 59, 547, 61, 31237, 37, 2, 1129, 2311

Offset: 1

Author

Labos Elemer, Jun 16 2003

Keywords

• base
• nonn

Comments

Conjecture: fixed point always exists.

Some initial values capriciously provide very large prime fixed-points. This behavior is illustrated in A084319 for initial value n=91.

Unlike the related home primes A037274, the trajectory of numbers in this procedure is not strictly increasing. Of the 8770 numbers < 10000 that have trajectories (that is, that are neither 1 nor prime) 3727 decrease at least once before reaching 30 digits. A sequence with no decreases is twice as likely to not terminate before 30 digits (10.0%) as one that has at least one decrease (4.8%). - Christian N. K. Anderson, May 04 2013

Examples

			a(0)=0 since no prime factors to concatenate;
a[p^j]=p for p prime(powers);
n=95=519: fixed-point list is {95,519,3173,19167,36389},
so a(95)=36389, a prime.
		

Links

• Christian N. K. Anderson, Table of n, steps to reach a(n), and a(n) for numbers where a(n) has fewer than 30 digits; NA otherwise. Also includes trajectories, factors separated by |s.

Crossrefs

Cf. A084317, A084319.

Programs

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] conc[x_] := Fold[nd, 0, Flatten[IntegerDigits[ba[x]], 1]] Table[FixedPoint[conc, w], {w, 1, 90}] Table[conc[w], {w, 1, 128}]