|
%I #30 Jan 02 2023 12:30:52
%S 190319129,191419228,192519327,193619426,194719525,195819624,
%T 196919723,290429139,291529238,292629337,293729436,294829535,
%U 295929634,390539149,391639248,392739347,393839446,394939545,490649159,491749258,492849357,493949456,590759169
%N Numbers N which are the concatenation of (a, a+b, b) in more than one way; a, b > 0.
%C If we allow b=0 then the first solutions are N = 11110, 12120, 13130, ..., i.e., concat(10x+y, 10x+y,0) = concat(x, 10y+x, 10y) with x, y in {1, ..., 9}.
%H M. F. Hasler and Hans Havermann (M. F. Hasler to 488), <a href="/A267770/b267770.txt">Table of n, a(n) for n = 1..2891</a>
%H E. Angelini, <a href="http://list.seqfan.eu/oldermail/seqfan/2016-February/016105.html">770700</a>, SeqFan list, Feb. 13, 2016.
%e 190319129 = 1903.1912.9 = 190.319.129, where "." denotes concatenation and the middle term is the sum of the first and last term, 1912 = 1903 + 9, 319 = 190 + 129.
%e 191419228 = 1914.1922.8 = 191.419.228, etc.
%e There are 28 terms with 9 digits that are all of this type, which we call (4,4,1) & (3,3,3).
%e Then there are 208 terms with 11 digits, namely 64 of type (4,4,3) & (1,5,5), 72 of type (5,5,1) & (3,4,4), and 72 of type (5,5,1) & (4,4,3).
%e Then there are 252 terms with 12 digits, ranging from 18913.19012.99 = 1891.3190.1299 to 79799.79818.19 = 7979.9798.1819, all of this type (5,5,2) & (4,4,4).
%o (PARI) show(LCc)={TCc=10^LCc; for(Lb=1,LCc-1, Tb=10^Lb; for(Cc=TCc\10,TCc-1, for(b=Tb\10,Tb-1, Aa = Cc-b; Tc=1;Ta=10; while( Cc > Tc*=10, TC=TCc\Tc; Cc%Tc*10<Tc && next; cb = Cc%Tc * Tb + b; while(Aa\Ta+cb>(aC=Aa%Ta*TC +Cc\Tc) && Aa > Ta*=10,); Aa>Ta || next(2); Aa\Ta+cb == aC && (Aa%Ta*10>Ta) && print1(Aa,Cc,b", ")))))}
%o for(L=1,5,show(L)) \\ Yields results only from n=4 on. WARNING, beyond the 29th term the list is no longer in order, some terms with LCc=5 are smaller than other terms with LCc=4.
%K nonn,base
%O 1,1
%A _Eric Angelini_, _Hans Havermann_ and _M. F. Hasler_, Feb 14 2016
|
