Comment by Erik Huang
A007770 Happy Numbers:
A happy number is defined by the following process: Starting with any positive integer,
replace the number by the sum of the squares of its digits in base-ten,
and repeat the process until the number either equals 1 (where it will stay),
or it loops endlessly in a cycle that does not include 1. Those numbers for which this
process ends in 1 are happy numbers, while those that do not end in 1 are unhappy numbers (or sad numbers).
GP code:
ssd(n)=n=digits(n); sum(i=1, #n, n[i]^2)
isA006753(n)=while(n>6, n=ssd(n)); n==1
for(i=1, 1e6, isA006753(i) && print (i))
There are infinitely many happy numbers and infinitely many unhappy numbers.
Consider following proof:
Let n be a happy number. Then 10n is also a happy number since
the sum of the squared digits of 10n equals to the sum of the
squared digits of n. Since 1 is a happy number, there are infinitely many happy numbers.
Using the same argument, applied to the unhappy number 2, we may
conclude there there are infinitely many unhappy numbers.