# forthcoming improvements to OEIS

N. J. A. Sloane njas at research.att.com
Wed Jul 2 20:17:24 CEST 2008

```Numbers with strictly increasing positive prime exponents.

2, 3, 5, 7, 11, 13, 17, 18, 19, 23, 29, 31, 37, 41, 43, 47, 50, 53,
54, 59, 61, 67, 71, 73, 79, 83, 89, 97, 98, 101, 103, 107, 108, 109,
113, 127, 131, 137, 139, 149, 151, 157, 162, 163, 167, 173, 179, 181,
191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263,
269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 338

This is a kind of inverse of A087980  Numbers with strictly decreasing
prime exponents.

Formula:
The numbers of the form Product[i=1..n] p_i^k_i where p_i =
A000040(i) is the i-th prime, and k_1 < k_2 < ... < k_n are positive
natural numbers.

A000040 UNION {2^i * p_j^k for i>0, j>2, k>i}={18, 50, 54, 98, 108,
162, 242, 338, ...} UNION {2^i * p_j^L * p_k^m for i>0, j>2, k>j, L>i,
m>L}={2^1 3^2 5^3 = 250, ...} UNION ...

Unless I've made a computation or search error, {18, 50, 54, 98, 108,
162, 242, 338, ...} is not yet in OEIS, nor the sequence initially
suggested.

There is some ambiguity in the definition, which first comes up in
asking: does 3^1 * 5^2 = 75 appear in the sequence?  If so, we lose
the primorial formulae analogues to A087980. That is, do we have every
consecutive prime as factors of the main sequence, or can we skip a
prime (as first comes up with 2^1 * (3^0) * 5^2 = 50, or with 3^1 *
(5^0) * 7^2 = 147)?  I may have erred in some inconsistency here.

Any thought on what, if anything, here is interesting?

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 should give A133811. For example 4=2^2 and 8=2^3 fulfill
the requirement. So is the (tacit) intend to skip prime powers?

The actual problem is that 50=2*5^2, 98=2*7^2, 147=3*7^2 and perhaps
others seem to be missing in A133811, which looks like an error to me and is
worth an independent check by someone else.

I get A133811 as follows:
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 A133811, we have
18,50,54,75,98,108,147,162,242,245,250,324,338
which probably is A097319...

Richard

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

```