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A328974 Trajectory of 1496 under repeated application of the map defined in A053392.

%I A328974 #28 Jan 10 2025 04:32:02
%S A328974 1496,51315,6446,10810,1891,91710,10881,18169,99715,181686,9971414,
%T A328974 18168555,99714131010,181685544111,99714131098522,18168554419171374,
%U A328974 99714131098510108841011,1816855441917136111816125112,99714131098510108849722997737623,1816855441917136111816121316941118161410101385
%N A328974 Trajectory of 1496 under repeated application of the map defined in A053392.
%C A328974 1496 is the smallest number whose trajectory under A053392 increases without limit.
%C A328974 More terms than usual are shown in order to display the onset of exponential growth.
%C A328974 Proof that this grows without limit, from _Hans Havermann_, Nov 01 2019: (Start)
%C A328974 One way to prove that the trajectory of a number under repeated application of the map defined in A053392 increases without limit is to show that there exists a term containing a non-final substring of three adjacent 9's.
%C A328974 If such a substring in term n is followed by a 0, it will grow to a non-final substring of three adjacent 9's followed by a 1 in term n+2: *9990* --> *18189* --> *99917*
%C A328974 If such a substring in term n is followed by a digit d that is neither 0 nor 9, it will grow to a non-final substring of four adjacent 9's in term n+2: *999d* --> *18181(d-1)* --> *9999d*
%C A328974 If such a substring in term n is followed by another 9, then it is a substring of k 9's for k >= 4, which will grow to a substring of (2k-3) 9's in term n+2: *[9]^d* --> *[18]^(d-1)* --> *[9]^(2d-3)*
%C A328974 Putting those together, such a substring in term n will grow to four adjacent 9's by term n+4; to five adjacent 9's by term n+6; to seven adjacent 9's by term n+8; ... to 2^k+3 adjacent 9's (see A062709) by term n+4+2k, regardless of what happens in the rest of the number.
%C A328974 In the present sequence a(41) contains two non-final substrings of three adjacent 9's. QED (End)
%C A328974 [Argument corrected and completed by _David J. Seal_, Nov 05 2019.]
%H A328974 Paolo Xausa, <a href="/A328974/b328974.txt">Table of n, a(n) for n = 1..32</a>
%H A328974 Erich Friedman, <a href="https://erich-friedman.github.io/mathmagic/0200.html">Problem of the Month (February 2000)</a>.
%t A328974 NestList[FromDigits[Flatten[{IntegerDigits[Total[Partition[IntegerDigits[#], 2, 1], {2}]]}]] &, 1496, 20] (* _Paolo Xausa_, Jan 10 2025 *)
%Y A328974 Cf. A053392.
%K A328974 nonn,base
%O A328974 1,1
%A A328974 _N. J. A. Sloane_, Nov 01 2019