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Old 2021-03-02, 20:25   #921
garambois
 
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"Garambois Jean-Luc"
Oct 2011
France

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New conjectures


We now have three lines of research in our project :

1) Work on the occurrences of prime numbers in the sequences and work on the prime numbers that terminate the sequences (resumption of the data analysis planned for the summer of 2021).
2) Work on the cycles that complete sequences, according to the bases (analysis work planned for a later date, because we have too few cycles for the moment).
3) Work on the increasing beginnings of sequences ending in a trivial way : new research axis which will be discussed in this post.

This new line of research was not part of the original idea of ​​the project and was discovered by chance.
This also happened for axis 2).

In the first part (I), I propose conjectures and I comment on these conjectures.
In a second part (II), I explain how I proceeded to observe the behaviors which made it possible to state the conjectures. I present a data table.
In a third part (III), I make remarks and I ask more general questions.


I) Statement of new conjectures :


I decided to number the conjectures in the continuity of those presented in the post #447.
So, we start here with the conjecture number (134).
The numbers we are talking about in this post are all integers.
The conjectures stated here concern sequences n^i which terminate trivially.
Reminder : This happens if n is a double of a square or if n and i have the same parity.
Preliminary remark : if the base is a prime number p, then s(n^p)<p.
For this reason, we only examine the growth of sequences from index 1.
We look at these sequences that must end trivially which first start growing from index 1 over one or more iterations.

Hope you find some interest in these new conjectures.
Some of these conjectures may already be known !

Some may also be invalidated in the future !

Conjecture (134) :
If a basis b = p# is primorial (p prime > 7), then the sequence that starts with the integer b^14 is increasing from index 1 for a few iterations.
Note 1 : In general, this does not seem to be the case with the other exponents, except exponent 8 (see conjecture 135).
Note 2 : We believe that if we compute the following larger primorial bases, the growth phenomenon will occur with other exponents. To check.
Note 3 : This conjecture is completed and replaced by conjecture (140).

Conjecture (135) :
If a basis b = p# is primorial (p prime > 29), then the sequence that starts with the integer b^8 is increasing from index 1 for a few iterations.
Note 1 : In general, this does not seem to be the case with the other exponents, except exponent 14 (see conjecture 134).
Note 2 : The same as for conjecture (134).
Note 3 : This conjecture is completed and replaced by conjecture (140).

Conjecture (136) :
If a base b = (p#) / 2 is primorial without the factor 2 (p prime> 3), then some sequences of this base grow from index 1 for a few iterations.
Note 1 : Until today (March 2021), we have only found three other odd bases for which this is also the case :
231 (3 * 7 * 11), 3003 (3 * 7 * 11 * 13) and 51051 (3 * 7 * 11 * 13 * 17),
to be seen as a primorial numbers without the factors 2 and 5 ?
Because 3003^5 and 51051^11 also have this property !
Note 2 : It is possible that this is just an illusion, maybe there are many other odd numbers that have the property ?
Note 3 : many of the exponents are prime numbers (especially 11 and 23) and not prime exponents are often equal to 7 * 5.

Conjecture (137) :
Base 2 sequences starting with 2^(12 * k), 2^(40 * k), 2^(90 * k), 2^(140 * k), 2^(210 * k), 2^(220 * k), 2^(330 * k), are increasing from index 1 for a few iterations.
Note 1 : This is not the case for the other exhibitors we have examined.
Note 2 : We believe that there must be other exponents of the form z * k (with z>330) which have this property.
Note 3 : We think that this phenomenon is related to the theorem which says that if p prime, s (p^i) is a factor of s (p^(i * m)) for every positive integer m, see post #466.
This theorem ensures, for example, that exponents multiple of 12 have many prime factors in their decomposition (like 2^12 itself), which ensures them growth for a few iterations.
Because of this mechanism, this conjecture looks more like those of post #447.


II) How to see these conjectures, with what data ?


To state these conjectures, it was necessary to present the data as clearly as possible, as below.
It looks very similar to the tables presented by Edwin, in the previous posts.
By looking at this data, anyone can discover their own conjectures.
Because it is certain that we missed things, perhaps even more spectacular or more general than what we present above.
It's your turn...


Code:
#########################################################################################
#####################   Warning :                                   #####################
#####################   We tested the odd exponents of numbers of   #####################
#####################   the form n = 2 * u^2 (u integer):           #####################
#####################   nothing at all !                            #####################
#########################################################################################

##########################################################################################
#####################   EVEN BASES, NUMBER OF BASES PROCESSED :42   ######################
##########################################################################################

******** Analysis base = 2 = 2,total number of exponents : 559, only 'even' exponents tested. ********

 2^12 growing from index 1 to 2  exponent factors : 2^2 * 3
 2^24 growing from index 1 to 2  exponent factors : 2^3 * 3
 2^36 growing from index 1 to 3  exponent factors : 2^2 * 3^2
 2^40 growing from index 1 to 2  exponent factors : 2^3 * 5
 2^48 growing from index 1 to 2  exponent factors : 2^4 * 3
 2^60 growing from index 1 to 2  exponent factors : 2^2 * 3 * 5
 2^72 growing from index 1 to 3  exponent factors : 2^3 * 3^2
 2^80 growing from index 1 to 2  exponent factors : 2^4 * 5
 2^84 growing from index 1 to 3  exponent factors : 2^2 * 3 * 7
 2^90 growing from index 1 to 2  exponent factors : 2 * 3^2 * 5
 2^96 growing from index 1 to 2  exponent factors : 2^5 * 3
 2^108 growing from index 1 to 4  exponent factors : 2^2 * 3^3
 2^120 growing from index 1 to 3  exponent factors : 2^3 * 3 * 5
 2^132 growing from index 1 to 7  exponent factors : 2^2 * 3 * 11
 2^140 growing from index 1 to 2  exponent factors : 2^2 * 5 * 7
 2^144 growing from index 1 to 3  exponent factors : 2^4 * 3^2
 2^156 growing from index 1 to 5  exponent factors : 2^2 * 3 * 13
 2^160 growing from index 1 to 2  exponent factors : 2^5 * 5
 2^168 growing from index 1 to 7  exponent factors : 2^3 * 3 * 7
 2^180 growing from index 1 to 5  exponent factors : 2^2 * 3^2 * 5
 2^192 growing from index 1 to 2  exponent factors : 2^6 * 3
 2^200 growing from index 1 to 2  exponent factors : 2^3 * 5^2
 2^204 growing from index 1 to 5  exponent factors : 2^2 * 3 * 17
 2^210 growing from index 1 to 2  exponent factors : 2 * 3 * 5 * 7
 2^216 growing from index 1 to 3  exponent factors : 2^3 * 3^3
 2^220 growing from index 1 to 2  exponent factors : 2^2 * 5 * 11
 2^228 growing from index 1 to 4  exponent factors : 2^2 * 3 * 19
 2^240 growing from index 1 to 3  exponent factors : 2^4 * 3 * 5
 2^252 growing from index 1 to 4  exponent factors : 2^2 * 3^2 * 7
 2^264 growing from index 1 to 5  exponent factors : 2^3 * 3 * 11
 2^270 growing from index 1 to 2  exponent factors : 2 * 3^3 * 5
 2^276 growing from index 1 to 4  exponent factors : 2^2 * 3 * 23
 2^280 growing from index 1 to 2  exponent factors : 2^3 * 5 * 7
 2^288 growing from index 1 to 4  exponent factors : 2^5 * 3^2
 2^300 growing from index 1 to 2  exponent factors : 2^2 * 3 * 5^2
 2^312 growing from index 1 to 6  exponent factors : 2^3 * 3 * 13
 2^320 growing from index 1 to 2  exponent factors : 2^6 * 5
 2^324 growing from index 1 to 3  exponent factors : 2^2 * 3^4
 2^330 growing from index 1 to 2  exponent factors : 2 * 3 * 5 * 11
 2^336 growing from index 1 to 8  exponent factors : 2^4 * 3 * 7
 2^348 growing from index 1 to 4  exponent factors : 2^2 * 3 * 29
 2^360 growing from index 1 to 5  exponent factors : 2^3 * 3^2 * 5
 2^372 growing from index 1 to 6  exponent factors : 2^2 * 3 * 31
 2^384 growing from index 1 to 5  exponent factors : 2^7 * 3
 2^396 growing from index 1 to 8  exponent factors : 2^2 * 3^2 * 11
 2^400 growing from index 1 to 2  exponent factors : 2^4 * 5^2
 2^408 growing from index 1 to 3  exponent factors : 2^3 * 3 * 17
 2^420 growing from index 1 to 5  exponent factors : 2^2 * 3 * 5 * 7
 2^432 growing from index 1 to 3  exponent factors : 2^4 * 3^3
 2^440 growing from index 1 to 2  exponent factors : 2^3 * 5 * 11
 2^444 growing from index 1 to 8  exponent factors : 2^2 * 3 * 37
 2^450 growing from index 1 to 2  exponent factors : 2 * 3^2 * 5^2
 2^456 growing from index 1 to 4  exponent factors : 2^3 * 3 * 19
 2^468 growing from index 1 to 4  exponent factors : 2^2 * 3^2 * 13
 2^480 growing from index 1 to 3  exponent factors : 2^5 * 3 * 5
 2^492 growing from index 1 to 7  exponent factors : 2^2 * 3 * 41
 2^504 growing from index 1 to 4  exponent factors : 2^3 * 3^2 * 7
 2^516 growing from index 1 to 4  exponent factors : 2^2 * 3 * 43
 2^520 growing from index 1 to 2  exponent factors : 2^3 * 5 * 13
 2^528 growing from index 1 to 4  exponent factors : 2^4 * 3 * 11
 2^540 growing from index 1 to 7  exponent factors : 2^2 * 3^3 * 5
 2^552 growing from index 1 to 10  exponent factors : 2^3 * 3 * 23

******** Analysis base = 6 = 2 * 3,total number of exponents : 206, only 'even' exponents tested. ********


******** Analysis base = 10 = 2 * 5,total number of exponents : 160, only 'even' exponents tested. ********

 10^36 growing from index 1 to 2  exponent factors : 2^2 * 3^2
 10^60 growing from index 1 to 2  exponent factors : 2^2 * 3 * 5
 10^72 growing from index 1 to 2  exponent factors : 2^3 * 3^2
 10^84 growing from index 1 to 2  exponent factors : 2^2 * 3 * 7
 10^90 growing from index 1 to 2  exponent factors : 2 * 3^2 * 5
 10^96 growing from index 1 to 4  exponent factors : 2^5 * 3
 10^108 growing from index 1 to 2  exponent factors : 2^2 * 3^3
 10^120 growing from index 1 to 3  exponent factors : 2^3 * 3 * 5
 10^132 growing from index 1 to 2  exponent factors : 2^2 * 3 * 11
 10^144 growing from index 1 to 3  exponent factors : 2^4 * 3^2

******** Analysis base = 12 = 2^2 * 3,total number of exponents : 148, only 'even' exponents tested. ********


******** Analysis base = 14 = 2 * 7,total number of exponents : 140, only 'even' exponents tested. ********

 14^36 growing from index 1 to 2  exponent factors : 2^2 * 3^2
 14^60 growing from index 1 to 2  exponent factors : 2^2 * 3 * 5
 14^72 growing from index 1 to 2  exponent factors : 2^3 * 3^2
 14^108 growing from index 1 to 2  exponent factors : 2^2 * 3^3
 14^120 growing from index 1 to 2  exponent factors : 2^3 * 3 * 5

******** Analysis base = 18 = 2 * 3^2,total number of exponents : 140, only 'even' exponents tested. ********

 18^120 growing from index 1 to 2  exponent factors : 2^3 * 3 * 5

******** Analysis base = 20 = 2^2 * 5,total number of exponents : 100, only 'even' exponents tested. ********

 20^36 growing from index 1 to 2  exponent factors : 2^2 * 3^2
 20^42 growing from index 1 to 2  exponent factors : 2 * 3 * 7
 20^60 growing from index 1 to 3  exponent factors : 2^2 * 3 * 5
 20^66 growing from index 1 to 2  exponent factors : 2 * 3 * 11
 20^72 growing from index 1 to 3  exponent factors : 2^3 * 3^2
 20^90 growing from index 1 to 2  exponent factors : 2 * 3^2 * 5
 20^96 growing from index 1 to 2  exponent factors : 2^5 * 3

******** Analysis base = 22 = 2 * 11,total number of exponents : 100, only 'even' exponents tested. ********

 22^6 growing from index 1 to 3  exponent factors : 2 * 3
 22^18 growing from index 1 to 2  exponent factors : 2 * 3^2
 22^60 growing from index 1 to 3  exponent factors : 2^2 * 3 * 5
 22^66 growing from index 1 to 3  exponent factors : 2 * 3 * 11
 22^100 growing from index 1 to 2  exponent factors : 2^2 * 5^2

******** Analysis base = 24 = 2^3 * 3,total number of exponents : 86, only 'even' exponents tested. ********


******** Analysis base = 26 = 2 * 13,total number of exponents : 100, only 'even' exponents tested. ********

 26^12 growing from index 1 to 2  exponent factors : 2^2 * 3
 26^18 growing from index 1 to 2  exponent factors : 2 * 3^2
 26^30 growing from index 1 to 2  exponent factors : 2 * 3 * 5
 26^36 growing from index 1 to 3  exponent factors : 2^2 * 3^2
 26^42 growing from index 1 to 2  exponent factors : 2 * 3 * 7
 26^54 growing from index 1 to 4  exponent factors : 2 * 3^3
 26^60 growing from index 1 to 2  exponent factors : 2^2 * 3 * 5
 26^72 growing from index 1 to 2  exponent factors : 2^3 * 3^2
 26^90 growing from index 1 to 3  exponent factors : 2 * 3^2 * 5
 26^96 growing from index 1 to 2  exponent factors : 2^5 * 3

******** Analysis base = 28 = 2^2 * 7,total number of exponents : 100, only 'even' exponents tested. ********

 28^36 growing from index 1 to 3  exponent factors : 2^2 * 3^2
 28^60 growing from index 1 to 2  exponent factors : 2^2 * 3 * 5
 28^72 growing from index 1 to 3  exponent factors : 2^3 * 3^2

******** Analysis base = 30 = 2 * 3 * 5,total number of exponents : 100, only 'even' exponents tested. ********


******** Analysis base = 34 = 2 * 17,total number of exponents : 100, only 'even' exponents tested. ********

 34^12 growing from index 1 to 2  exponent factors : 2^2 * 3
 34^24 growing from index 1 to 2  exponent factors : 2^3 * 3
 34^36 growing from index 1 to 3  exponent factors : 2^2 * 3^2
 34^48 growing from index 1 to 3  exponent factors : 2^4 * 3
 34^60 growing from index 1 to 4  exponent factors : 2^2 * 3 * 5
 34^72 growing from index 1 to 3  exponent factors : 2^3 * 3^2
 34^84 growing from index 1 to 3  exponent factors : 2^2 * 3 * 7
 34^90 growing from index 1 to 2  exponent factors : 2 * 3^2 * 5
 34^96 growing from index 1 to 3  exponent factors : 2^5 * 3

******** Analysis base = 42 = 2 * 3 * 7,total number of exponents : 100, only 'even' exponents tested. ********

 42^20 growing from index 1 to 2  exponent factors : 2^2 * 5
 42^32 growing from index 1 to 2  exponent factors : 2^5
 42^44 growing from index 1 to 2  exponent factors : 2^2 * 11
 42^56 growing from index 1 to 2  exponent factors : 2^3 * 7
 42^80 growing from index 1 to 4  exponent factors : 2^4 * 5

******** Analysis base = 50 = 2 * 5^2,total number of exponents : 95, only 'even' exponents tested. ********

 50^60 growing from index 1 to 3  exponent factors : 2^2 * 3 * 5
 50^72 growing from index 1 to 2  exponent factors : 2^3 * 3^2
 50^90 growing from index 1 to 2  exponent factors : 2 * 3^2 * 5

******** Analysis base = 72 = 2^3 * 3^2,total number of exponents : 88, only 'even' exponents tested. ********


******** Analysis base = 98 = 2 * 7^2,total number of exponents : 82, only 'even' exponents tested. ********

 98^36 growing from index 1 to 4  exponent factors : 2^2 * 3^2
 98^60 growing from index 1 to 2  exponent factors : 2^2 * 3 * 5
 98^72 growing from index 1 to 4  exponent factors : 2^3 * 3^2

******** Analysis base = 162 = 2 * 3^4,total number of exponents : 72, only 'even' exponents tested. ********


******** Analysis base = 200 = 2^3 * 5^2,total number of exponents : 80, only 'even' exponents tested. ********

 200^36 growing from index 1 to 2  exponent factors : 2^2 * 3^2
 200^60 growing from index 1 to 2  exponent factors : 2^2 * 3 * 5

******** Analysis base = 210 = 2 * 3 * 5 * 7,total number of exponents : 52, only 'even' exponents tested. ********


******** Analysis base = 220 = 2^2 * 5 * 11,total number of exponents : 80, only 'even' exponents tested. ********

 220^24 growing from index 1 to 3  exponent factors : 2^3 * 3
 220^54 growing from index 1 to 7  exponent factors : 2 * 3^3
 220^72 growing from index 1 to 2  exponent factors : 2^3 * 3^2

******** Analysis base = 242 = 2 * 11^2,total number of exponents : 52, only 'even' exponents tested. ********

 242^18 growing from index 1 to 3  exponent factors : 2 * 3^2
 242^36 growing from index 1 to 2  exponent factors : 2^2 * 3^2

******** Analysis base = 284 = 2^2 * 71,total number of exponents : 70, only 'even' exponents tested. ********

 284^30 growing from index 1 to 2  exponent factors : 2 * 3 * 5
 284^60 growing from index 1 to 2  exponent factors : 2^2 * 3 * 5

******** Analysis base = 288 = 2^5 * 3^2,total number of exponents : 50, only 'even' exponents tested. ********


******** Analysis base = 338 = 2 * 13^2,total number of exponents : 50, only 'even' exponents tested. ********

 338^24 growing from index 1 to 2  exponent factors : 2^3 * 3
 338^36 growing from index 1 to 2  exponent factors : 2^2 * 3^2
 338^48 growing from index 1 to 2  exponent factors : 2^4 * 3

******** Analysis base = 392 = 2^3 * 7^2,total number of exponents : 52, only 'even' exponents tested. ********

 392^36 growing from index 1 to 3  exponent factors : 2^2 * 3^2
 392^48 growing from index 1 to 2  exponent factors : 2^4 * 3

******** Analysis base = 450 = 2 * 3^2 * 5^2,total number of exponents : 50, only 'even' exponents tested. ********


******** Analysis base = 496 = 2^4 * 31,total number of exponents : 60, only 'even' exponents tested. ********

 496^30 growing from index 1 to 2  exponent factors : 2 * 3 * 5
 496^36 growing from index 1 to 3  exponent factors : 2^2 * 3^2

******** Analysis base = 578 = 2 * 17^2,total number of exponents : 50, only 'even' exponents tested. ********

 578^12 growing from index 1 to 2  exponent factors : 2^2 * 3
 578^16 growing from index 1 to 2  exponent factors : 2^4
 578^24 growing from index 1 to 3  exponent factors : 2^3 * 3
 578^36 growing from index 1 to 3  exponent factors : 2^2 * 3^2
 578^40 growing from index 1 to 4  exponent factors : 2^3 * 5
 578^48 growing from index 1 to 5  exponent factors : 2^4 * 3

******** Analysis base = 770 = 2 * 5 * 7 * 11,total number of exponents : 60, only 'even' exponents tested. ********

 770^54 growing from index 1 to 2  exponent factors : 2 * 3^3

******** Analysis base = 882 = 2 * 3^2 * 7^2,total number of exponents : 40, only 'even' exponents tested. ********

 882^40 growing from index 1 to 3  exponent factors : 2^3 * 5

******** Analysis base = 2310 = 2 * 3 * 5 * 7 * 11,total number of exponents : 35, only 'even' exponents tested. ********

 2310^14 growing from index 1 to 2  exponent factors : 2 * 7

******** Analysis base = 8128 = 2^6 * 127,total number of exponents : 40, only 'even' exponents tested. ********

 8128^36 growing from index 1 to 4  exponent factors : 2^2 * 3^2

******** Analysis base = 30030 = 2 * 3 * 5 * 7 * 11 * 13,total number of exponents : 26, only 'even' exponents tested. ********

 30030^14 growing from index 1 to 4  exponent factors : 2 * 7

******** Analysis base = 510510 = 2 * 3 * 5 * 7 * 11 * 13 * 17,total number of exponents : 22, only 'even' exponents tested. ********

 510510^14 growing from index 1 to 5  exponent factors : 2 * 7

******** Analysis base = 9699690 = 2 * 3 * 5 * 7 * 11 * 13 * 17 * 19,total number of exponents : 20, only 'even' exponents tested. ********

 9699690^14 growing from index 1 to 3  exponent factors : 2 * 7

******** Analysis base = 33550336 = 2^12 * 8191,total number of exponents : 25, only 'even' exponents tested. ********


******** Analysis base = 223092870 = 2 * 3 * 5 * 7 * 11 * 13 * 17 * 19 * 23,total number of exponents : 20, only 'even' exponents tested. ********

 223092870^14 growing from index 1 to 3  exponent factors : 2 * 7

******** Analysis base = 6469693230 = 2 * 3 * 5 * 7 * 11 * 13 * 17 * 19 * 23 * 29,total number of exponents : 14, only 'even' exponents tested. ********

 6469693230^14 growing from index 1 to 2  exponent factors : 2 * 7

******** Analysis base = 8589869056 = 2^16 * 131071,total number of exponents : 20, only 'even' exponents tested. ********


******** Analysis base = 200560490130 = 2 * 3 * 5 * 7 * 11 * 13 * 17 * 19 * 23 * 29 * 31,total number of exponents : 14, only 'even' exponents tested. ********

 200560490130^2 growing from index 1 to 3  exponent factors : 2
 200560490130^8 growing from index 1 to 3  exponent factors : 2^3
 200560490130^14 growing from index 1 to 2  exponent factors : 2 * 7

******** Analysis base = 7420738134810 = 2 * 3 * 5 * 7 * 11 * 13 * 17 * 19 * 23 * 29 * 31 * 37,total number of exponents : 14, only 'even' exponents tested. ********

 7420738134810^8 growing from index 1 to 2  exponent factors : 2^3
 7420738134810^14 growing from index 1 to 2  exponent factors : 2 * 7


#########################################################################################
#####################   ODD BASES, NUMBER OF BASES PROCESSED : 29   #####################
#########################################################################################


******** Analysis base = 3 = 3,total number of exponents : 335, only 'odd' exponents tested. ********


******** Analysis base = 5 = 5,total number of exponents : 250, only 'odd' exponents tested. ********


******** Analysis base = 7 = 7,total number of exponents : 189, only 'odd' exponents tested. ********


******** Analysis base = 11 = 11,total number of exponents : 153, only 'odd' exponents tested. ********


******** Analysis base = 13 = 13,total number of exponents : 145, only 'odd' exponents tested. ********


******** Analysis base = 15 = 3 * 5,total number of exponents : 140, only 'odd' exponents tested. ********

 15^35 growing from index 1 to 3  exponent factors : 5 * 7

******** Analysis base = 17 = 17,total number of exponents : 140, only 'odd' exponents tested. ********


******** Analysis base = 19 = 19,total number of exponents : 140, only 'odd' exponents tested. ********


******** Analysis base = 21 = 3 * 7,total number of exponents : 100, only 'odd' exponents tested. ********


******** Analysis base = 23 = 23,total number of exponents : 100, only 'odd' exponents tested. ********


******** Analysis base = 29 = 29,total number of exponents : 100, only 'odd' exponents tested. ********


******** Analysis base = 31 = 31,total number of exponents : 100, only 'odd' exponents tested. ********


******** Analysis base = 33 = 3 * 11,total number of exponents : 100, only 'odd' exponents tested. ********


******** Analysis base = 35 = 5 * 7,total number of exponents : 100, only 'odd' exponents tested. ********


******** Analysis base = 37 = 37,total number of exponents : 100, only 'odd' exponents tested. ********


******** Analysis base = 41 = 41,total number of exponents : 100, only 'odd' exponents tested. ********


******** Analysis base = 43 = 43,total number of exponents : 77, only 'odd' exponents tested. ********


******** Analysis base = 45 = 3^2 * 5,total number of exponents : 100, only 'odd' exponents tested. ********


******** Analysis base = 47 = 47,total number of exponents : 75, only 'odd' exponents tested. ********


******** Analysis base = 75 = 3 * 5^2,total number of exponents : 100, only 'odd' exponents tested. ********


******** Analysis base = 79 = 79,total number of exponents : 95, only 'odd' exponents tested. ********


******** Analysis base = 105 = 3 * 5 * 7,total number of exponents : 90, only 'odd' exponents tested. ********

 105^11 growing from index 1 to 2  exponent factors : 11
 105^23 growing from index 1 to 3  exponent factors : 23
 105^35 growing from index 1 to 3  exponent factors : 5 * 7
 105^47 growing from index 1 to 2  exponent factors : 47
 105^59 growing from index 1 to 3  exponent factors : 59
 105^83 growing from index 1 to 2  exponent factors : 83

******** Analysis base = 231 = 3 * 7 * 11,total number of exponents : 80, only 'odd' exponents tested. ********

 231^17 growing from index 1 to 3  exponent factors : 17
 231^77 growing from index 1 to 2  exponent factors : 7 * 11

******** Analysis base = 385 = 5 * 7 * 11,total number of exponents : 49, only 'odd' exponents tested. ********


******** Analysis base = 439 = 439,total number of exponents : 60, only 'odd' exponents tested. ********


******** Analysis base = 1155 = 3 * 5 * 7 * 11,total number of exponents : 50, only 'odd' exponents tested. ********

 1155^11 growing from index 1 to 3  exponent factors : 11
 1155^23 growing from index 1 to 3  exponent factors : 23
 1155^29 growing from index 1 to 3  exponent factors : 29
 1155^35 growing from index 1 to 4  exponent factors : 5 * 7
 1155^47 growing from index 1 to 3  exponent factors : 47

******** Analysis base = 15015 = 3 * 5 * 7 * 11 * 13,total number of exponents : 40, only 'odd' exponents tested. ********

 15015^3 growing from index 1 to 2  exponent factors : 3
 15015^7 growing from index 1 to 3  exponent factors : 7
 15015^9 growing from index 1 to 3  exponent factors : 3^2
 15015^11 growing from index 1 to 4  exponent factors : 11
 15015^15 growing from index 1 to 2  exponent factors : 3 * 5
 15015^19 growing from index 1 to 2  exponent factors : 19
 15015^23 growing from index 1 to 2  exponent factors : 23
 15015^27 growing from index 1 to 5  exponent factors : 3^3
 15015^29 growing from index 1 to 3  exponent factors : 29
 15015^31 growing from index 1 to 2  exponent factors : 31

******** Analysis base = 82589933 = 82589933,total number of exponents : 17, only 'odd' exponents tested. ********


******** Analysis base = 10000000019 = 10000000019,total number of exponents : 15, only 'odd' exponents tested. ********
III) General remarks and questions


Remark 1 :
All bases of the form b = 2 * n^2 produce growth from index 1 only for even exponents.
Can we observe the growth for the bases of this form with odd exponents ?

Remark 2 :
For some even and odd bases, there seems to be a lot of sequences with the properties of growth from index 1.
For others, there seems to be none.
If we notice that many "big" bases have only one sequence having the property, keep in mind that we can not calculate a lot of sequences with big exponents for these "big" bases.
It is impossible today to formulate a conjecture specifying which even or odd bases would have the property of growth or not.
But we see the advantage of calculating at least up to index 2 all the sequences which have an exponent of the same parity as the bases.
And above all, we understand that it would be necessary to calculate further the bases less than 10 (3, 5, 6, 7) which do not have the property of growth, because they are the easiest to calculate with large exponents.
It is extremely curious that for the base 2 * 3 we have nothing while for 2 * 5, 2 * 7, 2 * 11, 2 * 13, 2 * 17 we have as many !
What about 2 * 19 and 2 * 23 and ... ?

Remark 3 :
One could write a multitude of conjectures similar to the conjecture (137) bearing on other bases.
For example, for base 10, we could write a conjecture like this :
Base 10 sequences starting with 10^(36 * k), 10^(60 * k) are increasing from index 1 for a few iterations.
And if we continued the calculations further with higher exponents, we would certainly add 10^(84 * k), 10^(90 * k) and 10^(96 * k) to our conjecture.
Ditto for bases 14, 20, 22, 26 ...
Everyone can observe for themselves what is happening for all these bases.
We did not find it necessary to formulate all these similar conjectures here.

Remark 4 :
Conjectures 134, 135, and 136 are baffling and seem of a fairly new kind, while Conjecture 137 seems to come close to what we already know.

Remark 5 :
We have failed to make a more general guess.
But there must be a lot of new things to note.
We must take a good look at this data and add more data to it !

Last fiddled with by EdH on 2021-04-18 at 12:52 Reason: Per request in post 1086, added Note 3 to conjectures 134 and 135.
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