A130792 Numbers k whose representation can be split in two parts which can be used as seeds for a Fibonacci-like sequence containing k itself.

14, 19, 28, 47, 61, 75, 122, 149, 183, 199, 244, 298, 305, 323, 366, 427, 488, 497, 549, 646, 795, 911, 969, 1292, 1301, 1499, 1822, 1999, 2087, 2602, 2733, 2998, 3089, 3248, 3379, 3644, 3903, 4555, 4997, 5204, 5466, 6178, 6377, 6496, 6505, 7288, 7806, 7995

Offset: 1

Author

Giovanni Resta, Aug 20 2007

Keywords

• base
• nonn

Comments

The 6 terms with two digits are also Keith numbers. There are 233 numbers below 10^6 in this sequence.

Contribution from Paolo P. Lava, Apr 18 2025: (Start)

If the number k is rewritten as the concatenation of a and b, the problem is to find an integer x such that k = a*F(x) + b*F(x+1), where F(x) is a Fibonacci number (see in Links file with values of k, a, b, x, for k<10^6).

All the listed numbers admit only one concatenation that, through the addition process, leads to themselves. Is there any number that admits more than one single concatenation?

Sequence is infinite. Let us consider the numbers 19, 199, 1999, 19...9 and let us divide them as (1, 9), (1, 99), (1, 999), (1, 9...9). In two steps we have the initial numbers back: 1 + 9 = 10 and 9 + 10 = 19; 1 + 99 = 100 and 99 + 100 = 199, etc. (End)

Examples

			122 can be split into 12 and 2 and the Fibonacci-like sequence: 12, 2, 14, 16, 30, 46, 76, 122, ... contains 122 itself.
		

Links

• Michel Marcus, Table of n, a(n) for n = 1..406 (first 200 terms from Paolo Lava)
• Paolo P. Lava, List of k, a, b, x

Crossrefs

Cf. A007629.

Programs

Mathematica

testQ[n_]:= Block[{x, y, z, p = 10, r = False}, While[p < n, x = Floor[n/p]; y = Mod[n, p]; While[y < n, z = x + y; x = y; y = z]; If[y == n, r = True; Break[]]; p *= 10]; r]; Select[Range[10^4],testQ]

PARI

isok(n) = {nb = #Str(n); for (i=1, nb-1, x = n\10^i; y = n - 10^i*x; ok = 0; while(!ok, z = x + y; if (z > n, ok = 1); if (z == n, return (1)); x = y; y = z;));} \\ Michel Marcus, Oct 08 2014