# [seqfan] A001273 Extension

Hans Havermann pxp at rogers.com
Sat May 1 23:27:25 CEST 2010

```I realized that my method for extending A176762 [see Re: A175388(?)]
is also applicable to A001273...

-a(7) mod 81 = 24 ->  a(8) =  3789*10^973-1
-a(8) mod 81 = 62 ->  a(9) = 78889*10^((3789*10^973-306)/81)-1

The comment for A176762 states: a(8)= 3788 followed by 973 9's, a(9)=
237 followed by 4.6777777.......77777 x 10^974 9's - Sergio Pimentel
(ferdiego(AT)suddenlink.net), Feb 21 2007.  Sergio's a(9) is, I think,
238*10^((3789*10^973-63)/81)-1 which has one more digit than my a(9)
and would therefore be incorrect.

-a(9) mod 81 = 12 -> a(10) =   259*10^((78889*10^((3789*10^973-306)/
81)-94)/81)-1
a(10) mod 81 = 33 -> a(11) =
179*10^((259*10^((78889*10^((3789*10^973-306)/81)-94)/81)-115)/81)-1
a(11) mod 81 = 52 -> a(12) =
47*10^((179*10^((259*10^((78889*10^((3789*10^973-306)/81)-94)/81)-115)/
81)-53)/81)-1

Because this sequence is "also known as the smallest happy numbers of
height n", I was able to find an online reference:

http://rmmc.asu.edu/TO%20LINDA/tianx/happyNumbersRefsVersion.pdf

that not only reproduces my empirically-derived table (in a much more
rigorous fashion) on pages 4-5, but also verifies the above extension
numbers, albeit in a slightly altered format.

```