id: 06584246
dt: j
an: 2016c.00660
au: Schiffman, Jay L.
ti: Exploring the curiously fascinating integer sequence $1$, $12$, $123$,
$1234$, $12345$, $123456$, $1234567$, $12345678$, $123456789$,
$1234567891$, $12345678912$, $123456789123$,\dots.
so: Math. Spectr. 48, No. 2, 82-87 (2016).
py: 2016
pu: Applied Probability Trust (APT) c/o University of Sheffield, School of
Mathematics and Statistics (SoMaS), Sheffield
la: EN
cc: F60 I30
ut: sequences; factorization; divisibility
ci:
li:
ab: Summary: This article considers the integer sequence $1$, $12$, $123$,
$1234$, $12345$, $123456$, $1234567$, $12345678$, $123456789$,
$1234567891$, $12345678912$, $123456789123$,\dots . Our goal is to
examine the structure of the sequence by exploring divisibility
patterns including securing prime outputs and determining the highest
power of two that is a possible factor of any term in the sequence.
Using MATHEMATlCA$^\circledR$, I was able to obtain the complete prime
factorizations for the initial 108 terms in the sequence. The
deployment of modular arithmetic will enable us to secure recurring
prime factors from complete groupings such as $123456789$,
$123456789123456789$, $123456789123456789123456789$,\dots . We conclude
by suggesting future directions for companion sequences that serve to
furnish additional stimulating research. Such possibilities include
extensions, the sequence reversal, and examining the sequence and its
reversals in different number bases such as hexadecimal and duodecimal
(base twelve).
rv: