A243846 Numbers for which the nozero power-sequence of n falls into a loop.

1, 366784, 14877, 531136, 29287878125, 13631616, 18916327, 1245376, 118971, 1, 24871, 1942272, 377414623, 361123756, 221285675921484375, 453559756, 16185473, 4136832, 113758939, 366784, 164961711, 3179798512, 131147731, 1841716224, 283439365914625, 118754727776

Offset: 1

Author

Anthony Sand, Jun 12 2014

Keywords

• nonn
• base

Comments

Numbers returned by the following procedure: For n = 1, 2, 3, ..., let x(n; 1) = 1. Begin the recursive sequence x(n; i) = nozero(x(n; i-1) * n), where the function nozero(x) removes all zeros from x (see A004719). When x(n; i) = x(n; j

a(10*n) = a(n). - Pontus von Brömssen, May 19 2019

Examples

			a(2) = 366784 because x(2; 491) = nozero(183392 * 2) = 366784. Subsequently x(2; 527) = nozero(1533392 * 2) = nozero(3066784) = 366784, and this happens for the first time. Therefore x(2; 527) = x(2; 491) and the procedure returns x(2; 527) = 366784.
a(3) = 14877 because x(3; 28) = nozero(469359 * 3) = nozero(1408077) = 14877. Subsequently, x(3; 108) = nozero(4959 * 3) = 14877, and this happens for the first time. Therefore x(3; 28) = x(3; 108) and the procedure returns x(3;108) = 14877.
		

Links

• Pontus von Brömssen, Table of n, a(n) for n = 1..128
• Sean A. Irvine, Java program (github)

Crossrefs

Cf. A004719, A242350, A243845, A306569.

Programs

Mathematica

a[n_] := Block[{h = <||>, t = n}, While[! KeyExistsQ[h,t], h[t]=0; t = FromDigits@ Select[ IntegerDigits[n t], # > 0 &]]; t]; Array[a, 20] (* Giovanni Resta, May 20 2019 *)

Formula

Recurrence: x(n; i) = nozero(x(n; i-1) * n), x(n; 1) = 1, i >= 2, with n >= 1. For example, for x(2;10) = 512 and nozero(512 * 2) = nozero(1024) = 124. Therefore x(2;11) = 124.

If the sequence {x(n; i)}_{i >= 1} becomes periodic at some entry x(n; j), that is if there exists a period length L(n) such that x(n; i + L(n)) = x(n; i) for i >= j then a(n) = x(n; j). If there is no such period length then one puts a(n) = 0.

Extensions

Edited: Comment, formula and example reformulated. - Wolfdieter Lang, Jul 13 2014

a(5), a(6), a(8), a(9) corrected by Pontus von Brömssen, May 19 2019

a(10)-a(26) from Giovanni Resta, May 20 2019

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