[seqfan] A sequence and array
Kimberling, Clark
ck6 at evansville.edu
Sat Mar 6 15:48:15 CET 2010
Seqfans,
Let a(n) be the number of positive integers k such that n+k divides n*k.
(Equivalently, 1/n + 1/k = 1/m for some integer m.) The first 30 terms
of a(n) are 0,1,1,2,1,3,1,3,2,4,1,5,1,4,3,4,1,6,1,6,4,4,1,8,2,4,3,6,1,10
.
It's easy to prove that a(n)=1 iff n is a prime. Let's form the array
whose ith row consists of all n such that a(n)=i, so that (row 1)=(the
primes).
Which of the following assertions are true?
Row i includes all (prime)^i.
Every positive integer >1 is in the array.
The numbers in column 1 are even.
Clark Kimberling
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