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A328974 Trajectory of 1496 under repeated application of the map defined in A053392. 2
1496, 51315, 6446, 10810, 1891, 91710, 10881, 18169, 99715, 181686, 9971414, 18168555, 99714131010, 181685544111, 99714131098522, 18168554419171374, 99714131098510108841011, 1816855441917136111816125112, 99714131098510108849722997737623, 1816855441917136111816121316941118161410101385 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

1496 is the smallest number whose trajectory under A053392 increases without limit.

More terms than usual are shown in order to display the onset of exponential growth.

Proof that this grows without limit, from Hans Havermann, Nov 01 2019: (Start)

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.

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*

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*

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)*

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.

In the present sequence a(41) contains two non-final substrings of three adjacent 9's. QED (End)

[Argument corrected and completed by David J. Seal, Nov 05 2019.]

LINKS

Table of n, a(n) for n=1..20.

Erich Friedman, Problem of the Month (February 2000).

CROSSREFS

Cf. A053392.

Sequence in context: A237968 A062912 A328975 * A236732 A184078 A098191

Adjacent sequences:  A328971 A328972 A328973 * A328975 A328976 A328977

KEYWORD

nonn,base

AUTHOR

N. J. A. Sloane, Nov 01 2019

STATUS

approved

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Last modified December 4 12:21 EST 2020. Contains 338923 sequences. (Running on oeis4.)