Integer Sequences Partition Positive Integers

Leroy Quet qq-quet at
Mon May 10 22:20:02 CEST 2004

I just posted the message below to sci.math.  I am crossposting to, however, because it directly involves sequences of positive 
(It is along the same lines as a message I posted here a couple weeks 


Here is a result which is pretty much the same result mentioned in

but involves integer sequences, not real functions.

Let {a(k)} be any sequence of positive integers, where
a(k+1) >= a(k) for each k.
(a(1) is the sequence's first term.)

For m = a positive integer,
let b(k) be the number of elements
among the first m terms of {a(k)} which are <= k.

So, every positive integer from 1 to (a(m)+m-1) occurs exactly once
either in the sequence

{a(k)+k-1}, for 1<=k<=m,

or in

{b(k)+k}, for 1<=k<= a(m)-1,

but not in both sequences.
(And, for the range of k's, only these integers occur.)

For example:
Let a(k) = p(k), the k_th prime.

Let b(k) = pi(k), the number of primes <= k.

So, if m is infinity, we have the sequences:

{p(k)+k-1}: 2, 4, 7, 10, 15, 18, 23,...
{pi(k)+k}:  1, 3, 5, 6, 8, 9, 11, 12, 13, 14, 16,...

Leroy Quet

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