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
>
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