Why isn't 0 in A034710?
N. J. A. Sloane
njas at research.att.com
Sat Sep 22 12:59:17 CEST 2007
specify "Positive numbers". That will exclude 0,
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From: "N. J. A. Sloane" <njas at research.att.com>
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To: seqfan at ext.jussieu.fr
Subject: question about a possible duplicate
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Date: Sat, 22 Sep 2007 17:05:53 -0400
From: "Alexander Povolotsky" <apovolot at gmail.com>
To: seqfan at ext.jussieu.fr
Subject: Composite Integer Sequences consisting of multiple interleaved primitive sequences
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Hello,
Below is my (unpublished so far in OEIS) analysis of cases for "failed"
"n"'s for the following description of my sequence:
a(n) = (n^1 + 1)*(n^2 + 2)*(n^3 + 3)*...*(n^k + k)/k!
(See my comment in A131509).
In other words below analysis is for the cases when int k doesn't belong to
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 17, 18, 19, 20, 21}.
This analysis shows that those "failed" "n"'s form integer sequences of
their own.
Depending on the value of "k" - those are either "simple" "linear type"
sequences or "interleaved" ones,
with several (two or more) "participating" sequences (each of "linear type"
too).
NB This analysis is based on Peter J.C. Moses's conducted computations re
above mentioned failures.
Computations showed that for
i=|| {n }Failures
------------------------------------------------------
14|| {2,9,16,23,30,37,51,58,65,72,79,86,100,...}
15|| {2,9,16,23,30,37,51,58,65,72,79,86,100,...}
16|| {2,9,16,23,30,37,51,58,65,72,79,86,100,...}
22|| {7,18,29,40,51,62,73,84,95,...}
23|| {7,18,29,40,51,62,73,84,95,...}
24|| {7,18,29,40,51,62,73,84,95,...}
25|| {2,7,12,18,22,27,29,37,40,47,51,52,62,72,73,77,84,87,95,97,...}
26|| {7,11,18,24,29,37,40,50,51,62,73,76,84,89,95,...}
My (AP) analysis of above computational results comes to the following
observations:
>Failures
>i =
>14 {2,9,16,23,30,37,51,58,65,72,79,86,100,...}
>15 ....
>16 ....
Above sequence of "n" failures could be described as:
n = 7k + 2; for k=0,1,3,...,12, !!14, (16?)
Note that Multiplier K for above is 7
> ...
>22 {7,18,29,40,51,62,73,84,95,...}
>23
>24
n = 11k + 7; k=0,1,...,?
Note that Multiplier K for above is 11
>25 {2,7,12,18,22,27,29,37,40,47,51,52,62,72,73,77,84,87,95,97,...}
Above one could be broken in two intertwained and partially joined
sequences:
k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
---------------------------------------------------------------------------
2, 12, 22, 27, 37, 47, 52, 72, 77, 87, 97
62,
7, 18, ,29 ,40, 51, 73, 84, 95,
----------------------------------------------------------------------------
of those two:
for the first the description is:
n = 2 + 5k till k <= 9 and then less strictly
for the second the description is:
n = 7 + 11k
Note that Multiplier K for above composite integer sequence (which yields
two primitive ones) is 55 = 5*11
>26 {7,11,18,24,29,37,40,50,51,62,73,76,84,89,95,...}
k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
-------------------------------------------------------------------
7, 18, 29, 40, 51, 62 73, 84, 95 -> step 11
11 24 37 50 76 89 -> step 13
Note that Multiplier K for above is 143 = 11*13
It is also likely to predict that "n" failure sequence for i=28
will be a "simple" sequence with the step = 13,
Further, it is also likely to predict that the "n" failure sequence from
i=29 through i=33 could possibly be an interleaved combination of two
sequences with steps 7 and 13, since 17*13=91
Also, since 1001 = 7 * 11 * 13
then the "n" failure sequence for i=34 could be combination of 3 interleaved
sequences with steps 7, 11 and 13
Could someone (if interested) verify my above stated predictions ?
Should I report this to OEIS (and if yes, how should I "package" such report
?) ?
Thanks,
Best Regards,
Alexander Povolotsky
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Hello,<br><br>Below is my (unpublished so far in OEIS) analysis of cases for "failed" "n"'s for the following description of my sequence:<br>a(n) = (n^1 + 1)*(n^2 + 2)*(n^3 + 3)*...*(n^k + k)/k! <br>
(See my comment in A131509).<br><br>In other words below analysis is for the cases when int k doesn't belong to <br>{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 17, 18, 19, 20, 21}.<br><br>This analysis shows that those "failed" "n"'s form integer sequences of their own.
<br>Depending on the value of "k" - those are either "simple" "linear type" sequences or "interleaved" ones, <br>with several (two or more) "participating" sequences (each of "linear type" too).
<br>NB This analysis is based on Peter J.C. Moses's conducted computations re above mentioned failures.<br>Computations showed that for <br>i=|| {n }Failures<br>------------------------------------------------------
<br>14|| {2,9,16,23,30,37,51,58,65,72,79,86,100,...}<br>15|| {2,9,16,23,30,37,51,58,65,72,79,86,100,...}<br>16|| {2,9,16,23,30,37,51,58,65,72,79,86,100,...}<br>22|| {7,18,29,40,51,62,73,84,95,...}<br>23|| {7,18,29,40,51,62,73,84,95,...}
<br>24|| {7,18,29,40,51,62,73,84,95,...}<br>25|| {2,7,12,18,22,27,29,37,40,47,51,52,62,72,73,77,84,87,95,97,...}<br>26|| {7,11,18,24,29,37,40,50,51,62,73,76,84,89,95,...}<br><br>My (AP) analysis of above computational results comes to the following observations:
<br><br>>Failures<br>>i =<br>>14 {2,9,16,23,30,37,51,58,65,72,79,86,100,...}<br>>15 ....<br>>16 ....<br><br>Above sequence of "n" failures could be described as:<br>n = 7k + 2; for k=0,1,3,...,12, !!14, (16?)
<br>Note that Multiplier K for above is 7<br><br>> ...<br>>22 {7,18,29,40,51,62,73,84,95,...}<br>>23<br>>24<br><br>n = 11k + 7; k=0,1,...,?<br>Note that Multiplier K for above is 11 <br><br>>25 {2,7,12,18,22,27,29,37,40,47,51,52,62,72,73,77,84,87,95,97,...}
<br><br>Above one could be broken in two intertwained and partially joined sequences: <br>k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20<br>---------------------------------------------------------------------------
<br> 2, 12, 22, 27, 37, 47, 52, 72, 77, 87, 97<br> 62,<br> 7, 18, ,29 ,40, 51, 73, 84, 95, <br>----------------------------------------------------------------------------
<br>of those two:<br>for the first the description is:<br><br>n = 2 + 5k till k <= 9 and then less strictly <br><br>for the second the description is:<br><br>n = 7 + 11k<br><br>Note that Multiplier K for above composite integer sequence (which yields two primitive ones) is 55 = 5*11
<br><br><br>>26 {7,11,18,24,29,37,40,50,51,62,73,76,84,89,95,...}<br><br>k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 <br>-------------------------------------------------------------------<br> 7, 18, 29, 40, 51, 62 73, 84, 95 -> step 11
<br> <br> 11 24 37 50 76 89 -> step 13<br><br>Note that Multiplier K for above is 143 = 11*13<br><br>It is also likely to predict that "n" failure sequence for i=28<br>
will be a "simple" sequence with the step = 13,<br><br>Further, it is also likely to predict that the "n" failure sequence from i=29 through i=33 could possibly be an interleaved combination of two sequences with steps 7 and 13, since 17*13=91
<br><br>Also, since 1001 = 7 * 11 * 13<br>then the "n" failure sequence for i=34 could be combination of 3 interleaved sequences with steps 7, 11 and 13<br><br>Could someone (if interested) verify my above stated predictions ?
<br><br>Should I report this to OEIS (and if yes, how should I "package" such report ?) ?<br><br>Thanks,<br>Best Regards,<br>Alexander Povolotsky<br>
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