arXiv subjects
Publications and source records attributed to Christina Eubanks-Turner.
An augmented generalized happy function $S_{[c,b]}$ maps a positive integer to the sum of the squares of its base $b$ digits plus $c$. In this paper, we study various properties of the fixed points of $S_{[c,b]}$; count the number of fixed points of $§_{[c,b]}$, for $b \geq 2$ and $0<c<3b-3$; and prove that, for each $b \geq 2$, there exist arbitrarily many consecutive values of $c$ for which $S_{[c,b]}$ has no fixed point.
An augmented happy function, $S_{[c,b]}$ maps a positive integer to the sum of the squares of its base-$b$ digits and a non-negative integer $c$. A positive integer $u$ is in a cycle of $S_{[c,b]}$ if, for some positive integer $k$, $S_{[c,b]}^k(u) = u$ and for positive integers $v$ and $w$, $v$ is $w$-attracted for $S_{[c,b]}$ if, for some non-negative integer $\ell$, $S_{[c,b]}^\ell(v) = w$. In this paper, we prove that for each $c\geq 0$ and $b \geq 2$, and for any $u$ in a cycle of $S_{[c,b]}$, (1) if $b$ is even, then there exist arbitrarily long sequences of consecutive $u$-attracted integers and (2) if $b$ is odd, then there exist arbitrarily long sequences of 2-consecutive $u$-attracted integers.