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%I #14 Jan 15 2023 18:03:49
%S 1,2,3,4,5,6,7,8,9,10,20,30,40,50,60,70,80,90,109,806,1023,1044,2005,
%T 2676,3066,3602,4051,6053,6246,8011,8349,9427,10022,10074,10587,13090,
%U 15031,16867,20088,20699,21698,23108,29986,30091,30306,32226,40022
%N Hypercomma numbers: n occurs in the sequence S[k+1]=S[k]+10*last_digit(S[k-1])+first_digit(S[k]) for each "legal" splitting n=concat(S[0],S[1]).
%C This subsequence of A166511 consists of the numbers which occur as term in the sequence S(a,b), defined by S[0]=a, S[1]=b, S[k+1]=S[k]+10*last_digit(S[k-1])+first_digit(S[k]), for any legal splitting n=concat(a,b).
%C "Legal" means that a and b have at least one digit each, and b has no leading zero(s) (unless b=0). See A166511 and A166512 for more information.
%C They are called hypercomma numbers because they are k-comma numbers (cf. A166507) with k as large as possible for the given number of (zero and nonzero) digits, or "phoenix" numbers because they can be cut into (two) pieces is any (legal) way and will be "reborn" as a whole out of the "pieces".
%H E. Angelini, <a href="http://www.cetteadressecomportecinquantesignes.com/Commatile.htm">k-comma numbers</a>, Oct. 2009.
%H E. Angelini, <a href="/A166507/a166507.pdf">k-comma numbers</a> [Cached copy, with permission]
%e There is no legal way to split the single-digit numbers 1,...,9, therefore they are included.
%e More generally, a k-comma number which has exactly k nonzero digits when the last digit is ignored, will be in this sequence: e.g., 2005 can only be cut as (200,5); 10022 can only be cut as (1002,2) and (100,22), and it is a 2-comma number (A166512).
%o (PARI) {for(n=1,1e5,/*is_A166508(n)=*/ n%100 & for(d=1,#Str(n)-1, my( a=n\10^d, b=n%10^d ); b<10^(d-1) & d>1 & next /* not legal */; while(n > b=10*(a%10)+b\10^(#Str(b)-1)+a=b,); b>n & next(2) /* bad */); print1(n", "))}
%Y Cf. A166507, A166511, A166512, A166513.
%K base,nonn
%O 1,2
%A _Eric Angelini_ and _M. F. Hasler_, Oct 29 2009
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