Extended Dataset Generated by the OEIS Integer Sequence A376895: Primes of the Form 3^k*k^3 + 2
Author
Paul F. Marrero Romero
Title
Extended Dataset Generated by the OEIS Integer Sequence A376895: Primes of the Form 3^k*k^3 + 2
Description
This integer sequence was registered and published in the On-Line Encyclopedia of Integer Sequences (OEIS.org) Database on October 08 - 2024, under the OEIS code: A376895.
Category
Working Material
Keywords
OEIS, Integer Sequences, Discrete Mathematics, Data, Mathematics, Dataset, Prime Numbers, A376895
URL
http://www.notebookarchive.org/2024-11-2dul5yj/
DOI
https://notebookarchive.org/2024-11-2dul5yj
Date Added
2024-11-05
Date Last Modified
2024-11-05
File Size
182.02 kilobytes
Supplements
Rights
CC BY 4.0
Download
Open in Wolfram Cloud
Extended Dataset Generated by the OEIS Integer Sequence A376895: PRIMES OF THE FORM 3^K*K^3 + 2
Extended Dataset Generated by the OEIS Integer Sequence A376895: PRIMES OF THE FORM 3^K*K^3 + 2
Paul F. Marrero Romero
Numerical Data produced by the sequence of primes of the form Evaluated in the range of: 0 ≲ n ≲ 10000 / n ∈ ..
a(n)=*+2.
n
3
3
n
Details
Details
This integer sequence was registered and published in the On-Line Encyclopedia of Integer Sequences (OEIS.org) Database on October 08 - 2024, under the OEIS code: A376895.
This sequence can be expressed with the help of a general formula that uses the sequence A366997 which are the integers n such thatis a prime number. This general formula is such that:
This sequence can be expressed with the help of a general formula that uses the sequence A366997 which are the integers n such that
a(n)=*+2
n
3
3
n
a(n)=*+2.
A366997
3
3
A366997
(
1
)Some interesting properties of this sequence are:
◼
The next term is too large to include in the standard OEIS format.
a(8)=~2.20847
143
x10
◼
a(9)=~8.66244.
153
x10
◼
a(10)=~9.21872.
433
x10
◼
About the Code
About the Code
About the Mathematica code utilized for reproducing the data on the On-Line Encyclopedia of Integer Sequences, we utilized the following code:
Select[Table[3^k*k^3+2, {k, 0, 1000}], PrimeQ]
Select[Table[3^k*k^3+2, {k, 0, 1000}], PrimeQ]
The previous code executed in 0.134384 seconds (Wolfram Cloud) and 0.171601 seconds (Personal Computer).
The execution times were measured using the Mathematica function “Timing[]”.
The execution times were measured using the Mathematica function “Timing[]”.
The dataset reported on this notebook was generated by the following Mathematica program:
Select[Table[3^k*k^3+2, {k, 0, 10000}], PrimeQ]
Select[Table[3^k*k^3+2, {k, 0, 10000}], PrimeQ]
Exceeded base plan execution time in Wolfram Cloud. On other side, 718.01s were necessary to execute this code (Personal Computer).
Data Definitions
Data Definitions
Mathematica code that computes the sequence of prime numbers of the form a(n)=n3*3n+2 ) for non-negative integers n such that n ≤ 10000.
Mathematica code that computes the sequence of prime numbers of the form ) for non-negative integers such that n ≤ 10000.
a(n)=*+2
n
3
3
n
n
In[]:=
DSA376895=Select[Table[3^k*k^3+2,{k,0,10000}],PrimeQ](*TheextendedDataset*)
Mathematica plot code demonstrating the behavior of the sequence when evaluated for a non-negative integer n such that n ≤ 170 (Just like the published data in the OEIS).
Mathematica plot code demonstrating the behavior of the sequence when evaluated for a non-negative integer n such that n ≤ 170 (Just like the published data in the OEIS).
In[]:=
PLOTA376895=ListPlotSelect[Table[3^k*k^3+2,{k,0,170}],PrimeQ],PlotLabelHoldForm[A376985],JoinedTrue,AxesLabel->"n","p=*+2",LabelStyle->Directive[Black,Bold]
n
3
3
n
Note: The last code is designed to plot the data within the same ranges that were previously published in the OEIS database, primarily for the purpose of facilitating time calculations. Should the intention be to plot the entirety of the dataset displayed in this notebook, it is recommended that the values in {k, 0, 170} be replaced with {k, 0, 10000}.
Primary Content
Primary Content
Warning: To accurately reproduce the following Data and Plot, execute all Input code lines stated in the Data Definitions prior.
In[]:=
DSA376895
Out[]=
{2,5,14348909,3502727633,150094635296999123,269211745384444720788843377,2075640621314051693456929619860436129299430333182575810508680776710025092954370975575949,220847341205998801119151801849533569863346657553281896945729305733734542414037062994028409085361886122356496953109950384816443366723659662537377,8662441712540038408951314465509496761663924929545157124622810167082940687295523808477754379640867360318811157238517693922380063481815382500267002005632843,92187294029044193280067871797385749412570354769677756579202125305770357213981395992417594484188615505616747392283493644837664924699578678350063804480307410893136475157251526655169698737691770207250086647678942777330670618214402124526860904721191557277552534562790186239689394307189071875334344344640916392684250492325702444800575433538732692305038336016670309753039238193105524664164564611911710194323541262079939413047075052451061939,30810898517527917947487467749900662329535693595032169027081638274754977542253783381276094589268548012067358385635578927206938230693597503035087698089814806414127210095355869157201999076122275813017377813651345764656742191655066410177453708942954726965106993513787912570580543519178046478187722530782207326180955693235984273081525353441927917332923660374540304295969103669693173579834003950764008988278746649835538422215676625906045957550406853171355998236461270264813497538395575272435320014145618405979270228136810965047535391028344090881616667747375149080459480076846188536939616309782539339282930384235283815114138931693406806521915067509683345433109147099824016700846621763281344440566030515391807677357898787008118154280690136120459795798258866406192001180999361742358971630237608810329213393709739988301346513754291728344364526252470299785538019522631567638063073244170268593581393166769613981975952813595580204387468419490222482645739213400679513505692833187593433563449922415839057307872949133909309192667114553660258521443134138823293547846412385475126230344155109169237748985819630578915993315824458364345482091070429058418156373827009229211157654758395663553122167288748726597790291501120249650660305963861943344675871700252458814305526809550722766115680370660439322985978932174858531479897542579589027658238339476293561325547233972424672377480531868709252879390162209166733317920236301538027253885735177609801754660775128381005304150479897422811825743558747411706788946507583534226779201376796972641387884341550368831003736099356107455597020213929336312470128960658271353973829236474365922323309459348588469765743325101987614682572235745962554774549763494456827895839869096193751249633217883881123925600459429884823805558577832557584278926189402824192176635116736170661618805955764438257418040090296006954676205871325131542095607485233849631520826210137808064082873200064651915178173896266612871333248831027667142238267298865091027802023477820588149141688631932769946476826350544809571703147166905775821231197080975925204421930512222909782262052580814903209829722610860457397128424187121058458659349689215739096298304247505252055018924407403833273734682524876246855877,331744494323508586752315527438344311106382656231332526413471631734415927862745577253154825275874166909661505401080572415040320736755921258345227354193518600343153535291585159059150475098011152350625951887265814819425328414390649468180567052458322867067943628241649118662667093571913084561382613589289465488415543234656735362218578521380495947660718640813443391579421062811288124881208055398348258856691210692568483757534486834610236841460715512531885332225713482481037466464836206315320192343201722568799991768372801162608715272308147419100143467187051479039176919048256545433312259135468005404769923663525942148596262213496625462377910375089738065597938744304424673448024594142793582586999473942429828900288453569910611846635522084471669988012405392336146331865296967381829335534568199077551933626904153387443264834111125097136850779826047883881035857103099595159173660052028291452403479613028787418918654278067122791469696584559093897654930322331520201720613587971565429482411091935519782368425658840499989707151382389105283352524764905456192908247549958794404045380002686656684549723157333412511798257551538754101030221338315383551940400423263723686204940975049755321504712276717174513137803329765941164604470725328590254537627122066044123992941232519172426115357121707548621937938938273509987814546561223044049899118371478951227933795120754955581811496140695231757979188697129695308509134158803874683755938302628678640548425607769614234492857595181834570962708068766791447613415444186103332540811600432517398023852189774182902074962620055669876998573073628829878126865376764570265602167405188178374931477973303627748033696187340050618274936433604227483111640657115354950129486755974458576167776445889645818160843545021375397566192313250528020886328201280708882559374993592132013692223789767978566715302045279557121503770188746557935517611140465631400642418011462307158719106906066635520848083200165836319756588350646287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Examples
Examples
For easier manipulation of the data, the contents of dataset DSA376895 can be expressed in scientific notation.
In[]:=
DSA376895//N
Out[]=
{2.,5.,1.43489×,3.50273×,1.50095×,2.69212×,2.07564×,2.20847×,8.66244×,9.21872940290442×,3.081089851752792×,3.317444943235086×}
7
10
9
10
17
10
26
10
87
10
143
10
153
10
433
10
2179
10
4388
10
The calculation of the additive digital root of prime numbers of the form *+2 can be done as follows:
n
3
3
n
In[]:=
Mod[DSA376895,9,1]
Out[]=
{2,5,2,2,2,2,2,2,2,2,2,2}
In[]:=
3^20803*20803^3+2//N
Out[]=
3.219881140655294×
9938
10
In[]:=
PLOTA376895
Out[]=
Source & Additional Information
Source & Additional Information
Submitted By
Submitted By
Paul F. Marrero Romero
Source/Reference Citation
Source/Reference Citation
Paul F. Marrero Romero, Primes of the form 3^k*k^3 + 2. , Entry A376895 in The On-Line Encyclopedia of Integer Sequences, https://oeis.org/A376895
Detailed Source Information
Detailed Source Information
Author/Creator
Author/Creator
Paul F. Marrero Romero
Source Title
Source Title
Primes of the form 3^k*k^3 + 2.
Source Date
Source Date
October 08 - 2024.
Source Publisher
Source Publisher
Geographic Coverage
Geographic Coverage
Worldwide
Source Language
Source Language
English
Links
Links
◼
Integer sequence:
◼
Integer sequence:
◼
Integer sequence:
Keywords
Keywords
◼
Sequences
◼
Mathematica
◼
Integer sequences
◼
OEIS
◼
Discrete Mathematics
◼
Prime Numbers
◼
Algorithm
Categories
Categories
Content Types
Content Types
Author Notes
Author Notes
This sequence is currently under our study, and it is our intention to publish a corresponding paper discussing its algebraic properties and other aspects of the corresponding mathematics. The sequence is the result of some of the research projects that we are conducting in the field of discrete mathematics in Marrero Research Lab.
Personal Computer specs:
◼
Intel(R) Core(TM) i3-4005U CPU @ 1.70GHz 1.70 GHz
◼
6,00 GB Ram - DDR4
◼
SO: Windows 10 x64 Professional.
Cite this as: Paul F. Marrero Romero, "Extended Dataset Generated by the OEIS Integer Sequence A376895: Primes of the Form 3^k*k^3 + 2" from the Notebook Archive (2024), https://notebookarchive.org/2024-11-2dul5yj
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