A248134 Consider a number x as a concatenation of two integers, a and b: x = concat(a,b). Take their sum and repeat the process deleting the minimum number and adding the previous sum. The sequence lists the numbers that after some iterations reach a sum equal to themselves.

14, 19, 21, 28, 42, 47, 63, 84, 105, 126, 147, 149, 168, 189, 199, 298, 323, 497, 646, 795, 911, 969, 1292, 1499, 1822, 1999, 2087, 2733, 2998, 3089, 3248, 3379, 3644, 4555, 4997, 5411, 5466, 6178, 6377, 6496, 7288, 7995, 8199, 9161, 9267, 9744, 10822, 12356

Offset

1,1

Comments

If the number x is rewritten as concat(a,b), the problem is to find a value of y such that x = a*F(y) + b*F(y+1), if a < b, or x = b*F(y) + a*F(y+1), if a > b, where F(y) is a Fibonacci number (see values of x, a, b, y, for 1<x<10^6, in Links).

Similar to A130792 but here the minimum number is deleted since the beginning.

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

Sequence is infinite. Let us consider the numbers 19, 199, 1999, 19...9 and let us divide them as concat(1,9), concat(1,99), concat(1,999), concat(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.

Links

Paolo P. Lava, Table of n, a(n) for n = 1..410

Paolo P. Lava, List of [x,a,b,y] in the equation x = a*F(y) + b*F(y+1), for 1< x <10^6

Example

Let us rewrite 5411 as 54 U 11. Then:
11 + 54 = 65;
54 + 65 =  119;
65 + 119 = 184;
119 + 184 = 303;
184 + 303 = 487;
303 + 487 = 790;
487 + 790 = 1277;
790 + 1277 = 2067;
1277 + 2067 = 3344;
2067 + 3344 = 5411, that is 11*F(10) + 54*F(11) = 11*55 + 54*89 = 605 + 4806 = 5411.

Maple

P:=proc(q, h) local a, b, k, n, t, v; v:=array(1..h);
for n from 1 to q do for k from 1 to ilog10(n) do
a:=n mod 10^k; b:=trunc(n/10^k); if a<b then
v[1]:=a; v[2]:=b; else v[1]:=b; v[2]:=a; fi; t:=3;
v[t]:=a+b; while v[t]<n do t:=t+1; v[t]:=v[t-2]+v[t-1]; od;
if v[t]=n then print(n); break; fi; od; od; end: P(10^9, 1000);

Crossrefs

Cf. A000045, A007629, A130792, A246544, A247012, A247013.

Sequence in context: A222576 A165228 A205866 * A013649 A013657 A013653

Adjacent sequences: A248131 A248132 A248133 * A248135 A248136 A248137

Keyword

nonn,base

Author

Paolo P. Lava, Oct 02 2014

Status

approved