Strings of Consecutive Happy Numbers

In Spring 2008, one of my senior seminar students inspired me to work on the problem of strings of consecutive happy numbers, eventually resulting in the paper

Robert Styer, Smallest Examples of Strings of Consecutive Happy Numbers, Journal of Integer Sequences, Vol. 13, 2010, Article 10.6.3.   http://www.cs.uwaterloo.ca/journals/JIS/vol13.html

In Spring 2012 another student (Daniel Lyons) extended these results to include the smallest string of 14 or 15 consecutive happy numbers.  Here are his results.

Here are some results:

The first sequence of six consecutive happy numbers happens to be the first sequence of seven consecutive happy numbers, namely, the sequence that begins with

N = 7899999999999959999999996

For eight, the smallest string begins with

N = 58 * 10^{157} + 10^{146} (10^{11}-1) + 6 *10^{145} + 10 (10^{144}-1) + 5
= 5899999999999699999999999999999999999999999999999999999999999999999999999/
99999999999999999999999999999999999999999999999999999999999999999999999999999999999995
=

that is, a 159 digit number given by the digits 58 followed by 11 digits nine, then the digit 6, then 144 digits nine, and ending with the digit 5.

The smallest starting number that gives a string of nine consecutive happy numbers has 215 digits:

N = 26 * 10^{213} + 10^{76} (10^{137}-1) + 7 * 10^{75} + 10 (10^{74}-1) + 5
= 2699999999999999999999999999999999999999999999999999999999999999999999999/
9999999999999999999999999999999999999999999999999999999999999999997/
999999999999999999999999999999999999999999999999999999999999999999999999995

while the smallest for ten consecutive values has 651 digits:
N = 38 * 10^{649} + 10^{89} (10^{560}-1) + 10(10^{87}-1) + 5

For eleven, a 1571 digit number:
N = 27*10^{1569}+(10^{280}-1)*10^{1289}+(10^{1287}-1)*10+4.

12 in row has 158162 digits    N = 388.(158021 nines).8.(136 nines).4
13 in row has 603699 digits    N = 288.(218491 nines).3.(385203 nines).3

Daniel Lyon's results:

14 in a row :   7888.(1604938271577 nines).1.(345696 nines).3

15 in a row:   N=77.(2222222222222220 nines).3.(97388 nines).3

Here is a list of the smallest starting number for several `3-consecutive' sequences of cubic happy numbers.

Length of the sequence          Smallest N
2                                        1198
3                                        169957
4                                        1555599999999916
5                                        35588899999799999999999999989
6                                        28888 * 10^{96}+ (10^{21}-1)* 10^{75}+1 *10^{74} + (10^{72}-1)* 10^{2} + 89
7                                        3577*10^{230}+(10^{228}-1)*10^2+45
8                                        1126* 10^{238} + (10^{229} -1)*10^9 + 199999989
9                                        12777*10^{271}+(10^{151}-1)*10^{120}+5*10^{119}+(10^{117}-1)*10^2+86

A draft of a paper containing these results is here.  The final paper is published: Smallest Examples of Strings of Consecutive Happy Numbers, Journal of Integer Sequences, Vol. 13, 2010, Article 10.6.3.  http://www.cs.uwaterloo.ca/journals/JIS/vol13.html

This paper refers to some Maple worksheets:
Finding six and seven consecutive happy numbers  mw   pdf
Finding eight consecutive happy numbers mw   pdf
Finding nine consecutive happy numbers  mw    pdf
Finding ten consecutive happy numbers  mw   pdf
Finding 11 in a row  mw   pdf
Finding 12 in a row  mw   pdf
Finding 13 in a row  mw pdf

Finding the best "ell" value of El-Sedy and Siksek   mw   pdf
Finding the minimal N such that S(N) = n for a given n     mw    pdf

For the cubic happy numbers:
Finding strings of four or five cubic happy numbers   mw   pdf
Finding strings of six cubic happy numbers   mw   pdf
Finding strings of seven to nine cubic happy numbers   mw   pdf
Finding the minimal N such that S_3(N) = n for a given n when S_3 is the cubic digit function   mw   pdf.

Styer Home Page           08/10/2012 