Sequences of Happy Numbers

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
= 589999999999969999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999995
=

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
= 26999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999997999999999999999999999999999999999999999999999999999999999999999999999999995

while the smallest for ten consecutive values has 651 digits:

N = 38 * 10^{649} + 10^{89} (10^{560}-1) + 10(10^{87}-1) + 5

The Maple worksheet containing the details is here, a pdf version here. 

This worksheet uses a function that gives the minimal N such that S(N) = n for a given n, with details in this Maple worksheet.