# Proposed seq: Numbers with strictly increasing positive prime exponents

Richard Mathar mathar at strw.leidenuniv.nl
Wed Jul 2 22:41:59 CEST 2008

```So the minimum change to OEIS would be a comment tightening or
disambiguating the definition, and cross references with at least:

A071365, A097318, A097319, A097320, A112769
as well as all that I previously mentioned?

A variation would be that one cannot skip any primes AFTER some
starting index.  That is: product of consecutive primes, each with
exponents strictly greater than the exponent of the previous prime.

For instance, skip p_1 = 2 and start with p_2 = 3 and we have:
{3^i, for i>0} which we can eliminate if we are removing prime powers;
UNION
{3^i * 5^j, for i>0, j>i}
UNION
{3^i * 5^j * 7^k, for i>0, j>i, k>j}
UNION
{3^i * 5^j * 7^k * 11^L, for i>0, j>i, k>j, L>k}
UNION ...

These are then factors of the sequences built from primorials as cited
in comments or formulae of A087980.

On 7/2/08, Richard Mathar <mathar at strw.leidenuniv.nl> wrote:
>
>  My previous e-mail was too quick in condemning A133811, unfortunately...
>
>  jvp> From seqfan-owner at ext.jussieu.fr  Wed Jul  2 21:58:29 2008
>  jvp> Date: Wed, 2 Jul 2008 12:52:40 -0700
>  jvp> From: "Jonathan Post" <jvospost3 at gmail.com>
>  jvp> To: SeqFan <seqfan at ext.jussieu.fr>
>  jvp> Subject: Proposed seq: Numbers with strictly increasing positive prime exponents
>  jvp>
>  jvp> Numbers with strictly increasing positive prime exponents.
>  jvp>
>  jvp> 2, 3, 5, 7, 11, 13, 17, 18, 19, 23, 29, 31, 37, 41, 43, 47, 50, 53,
>  jvp> 54, 59, 61, 67, 71, 73, 79, 83, 89, 97, 98, 101, 103, 107, 108, 109,
>  jvp> 113, 127, 131, 137, 139, 149, 151, 157, 162, 163, 167, 173, 179, 181,
>  jvp> 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263,
>  jvp> 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 338
>  jvp>
>  jvp> This is a kind of inverse of A087980  Numbers with strictly decreasing
>  jvp> prime exponents.
>  jvp>
>  jvp> Formula:
>  jvp>  The numbers of the form Product[i=1..n] p_i^k_i where p_i =
>  jvp> A000040(i) is the i-th prime, and k_1 < k_2 < ... < k_n are positive
>  jvp> natural numbers.
>
>  By definition, this is close to A133811. For example 4=2^2 and 8=2^3 fulfill
>  the requirement. So is the (tacit) intend to skip prime powers?
>
>  If one relaxes the requirement in A133811 that the numbers are primally tight,
>  we have
>
>  2,3,4,5,7,8,9,11,13,16,17,18,19,23,25,27,29,31,32,37,41,43,47,49,50,53,54,59,
>  61,64,67,71,73,75,79,81,83,89,97,98,101,103,107,108,109,113,121,125,127,128,
>  131,137,139,147,149,151,157,162,163,167,169,173,179,181,191,193,197,199,211,
>  223,227,229,233,239,241,242,243,245,250,251,256,257,263,269,271,277,281,283,
>  289,293,307,311,313,317,324,331,337,338
>  (the leading 1 is a matter of definition, I am willing to negotiate...)
>
>  jvp> A000040 UNION {2^i * p_j^k for i>0, j>2, k>i}={18, 50, 54, 98, 108,
>  jvp> 162, 242, 338, ...} UNION {2^i * p_j^L * p_k^m for i>0, j>2, k>j, L>i,
>  jvp> m>L}={2^1 3^2 5^3 = 250, ...} UNION ...
>  jvp>
>  jvp> Unless I've made a computation or search error, {18, 50, 54, 98, 108,
>  jvp> 162, 242, 338, ...} is not yet in OEIS, nor the sequence initially
>  jvp> suggested.
>  jvp>
>  jvp> There is some ambiguity in the definition, which first comes up in
>  jvp> asking: does 3^1 * 5^2 = 75 appear in the sequence?  If so, we lose
>  jvp> the primorial formulae analogues to A087980. That is, do we have every
>  jvp> consecutive prime as factors of the main sequence, or can we skip a
>  jvp> prime (as first comes up with 2^1 * (3^0) * 5^2 = 50, or with 3^1 *
>  jvp> (5^0) * 7^2 = 147)?  I may have erred in some inconsistency here.
>  jvp>
>  jvp> Any thought on what, if anything, here is interesting?
>
>  If prime powers are removed from the sequence above, we have
>  18,50,54,75,98,108,147,162,242,245,250,324,338
>  which probably is A097319...
>
>
>  Richard
>

```