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A267770
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Numbers N which are the concatenation of (a, a+b, b) in more than one way; a, b > 0.
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1
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190319129, 191419228, 192519327, 193619426, 194719525, 195819624, 196919723, 290429139, 291529238, 292629337, 293729436, 294829535, 295929634, 390539149, 391639248, 392739347, 393839446, 394939545, 490649159, 491749258, 492849357, 493949456, 590759169
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OFFSET
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1,1
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COMMENTS
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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}.
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LINKS
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E. Angelini, 770700, SeqFan list, Feb. 13, 2016.
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EXAMPLE
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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.
191419228 = 1914.1922.8 = 191.419.228, etc.
There are 28 terms with 9 digits that are all of this type, which we call (4,4,1) & (3,3,3).
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).
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).
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PROG
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(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", ")))))}
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.
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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