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User:Bill McEachen

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          • I am mostly offline for November 2023 *****

older archived version: [1]

Rsz mceachen profile.png

I am a control systems engineer (retired) living in Virginia, USA. Born in 1962.

Below are/will be images of some relevant edits.--Bill McEachen 15:02, 26 April 2017 (UTC)

In the following areas (author/comment/formula) my counts as of Feb 2022 are ~ 47, 62 and 14. I work nearly always with Pari/GP, though a few times with Python, R, GAP and Gnumeric. As ordered

This sequence has relationship to A001122. I checked a few locations in the sequence, to check if some constant akin to Artin's seemed to hold. It does, perhaps ~ 0.267+. I am unsure if this is some already established constant. Here is a log-log scatter plot for A269844 vs A001122.--Bill McEachen 01:44, 25 January 2018 (UTC)

Misc searches

comment:"is this true"
comment:conjecture AND **
comment:"it is likely"
comment:"it is unlikely"
comment: "remarkable"
comment:"is it possible"

Program languages seen in OEIS

I was just curious and used "program:xxx" to try to get approximate counts. What I found ~ Sep 2022 (note this excludes Mathematica and Maple):

Pari ~ 103K
Magma ~ 32877
Python ~ 12304
Sage ~ 10587
Haskell ~ 8180
Maxima 2648
Ruby 412
MatLab 310
Basic 157
MuPad 126
Fortran <100 (no sequences after 140xxx)
R unknown
C++ unknown

A few unaccepted contributions

4/3/2015 my comment for A205083 was rejected, despite being true. The comment was: The sequence is characterized by an expected mean of 0.5, though pair sums ratio to 1/2/2 for 0,1,2 respectively (unlike a coin flip, which yields 1/2/1).
11/2019 comment on A086788 was rejected (was not explained well/understood). The comment was something like: Define an ellipse with prime semiaxes p and q, with semi-major axis p evaluated over the primes, and semi-minor axis q evaluated over the primes q <= p. Terms are those p failing to yield an ellipse area whose nearest integer is prime.
The Pari script is: The Pari script is:

genit(maxx)={forprime(p=2,maxx,q=p+1;ok=0; while(q>2, q=precprime(q-1);cand=round( p*q*Pi );

if(ispseudoprime(cand),ok=1;break));if(!ok,cand=0;fcnt+=1); print(p," ",q," ", cand));print("done, fails = ",fcnt);} genit(200) 

I entered a Thue-Morse like sequence. Date TBD.

Misc Notes

Connection between Orderly numbers (A167408) and Ulam numbers (A002858) can be seen via A167408/a167408.jpg

Generating primes

See A242902 and some concatenation sequences( A240563, A226095 ).

Prime-heavy sequences

(NOT from obvious direct primality cause but as a consequence of something else). A271116/A128288/A270951/A270997/(A014657)

Common terms in Name field

I was curious and list some counts here (I have a much fuller list). Of course counts are dynamic, these are as of mid-Jan 2022

square 13827
diagonal 11398
sqrt 10237
partition 10415
n! 200359
/2 110910
expansion 27723
triangle 19598
binary 11755
difference 10574
least 12255
product 10501
<=n 200359
number 167302
length 14099

A bit fewer:

divisor 9517
permutation 8335
g.f: 9698

Bitwise operators

Many sequences can be tied to bitwise operators as seen in the table below. The same script was used, changing only the operator and which integer pair is operated upon. Comments made regarding these were not well received.

  pair form      operation      sequence     note
  (m, m+2)      bitwise OR       A002145     m>0, odd, 10K terms
  (m, m+2)      bitwise AND      A002144     m>0, odd, 10K terms
  (m, m+4)      bitwise OR       A003628     m>0, odd, 10K terms
  (m, m+4)      bitwise AND      A033200     m>0, odd, 10K terms
  (m, m+1)      bitwise NEG      A081296     m>0, even, <9 terms (n>1)
  (m, m+2)      bitwise NEG      A100362     m>0, even, <15 terms (n>2)
  (m, m+1)      bitwise XOR      (TBD)       m>0, even, ** terms

Here is a link to some of the proofs:[[2]] covers A002144,A002145,A003628,A033200 Here is a link to the A081296 bitwise proof: [[3]]

Deceased listing

via search on "passed away",

User:Klaus Brockhaus
User:Ant King
User:Terrel Trotter, Jr
User:Labos Elemer
User:Enoch Haga
User:Cino Hilliard
User:Dan Hoey
User:David S. Johnson
User:Donovan Johnson
User:Zerinvary Lajos
User:Parthasarathy Nambi
User:David Scambler
User:D n Verma
User:Barry E. Williams
User:Reinhard Zumkeller
User:J. H. Conway
User:Jonathan Sondow (category:Deceased users is missing )
User:Richard K. Guy
User:John W. Layman
User:Ki Punches
User:Herb Conn
User:Claude Lenormand
User:Vladimir Shevelev
User:Farideh Firoozbakht
User:Philippe Flajolet
User:John Riordan
Solomon Golomb, ??

Ongoing effort/submittals

Items that await more slots opening:

Bill McEachen (talk) 23:49, 9 August 2023 (EDT)


My experiences with chatGPT/Bard on select math queries. The conclusion is, as of when the queries were run, chatGPT cannot be relied upon certainly for math. (all were run prior to Apr 12 2023, pgm version Mar 23)
The following queries were run, all with something incorrect in the subsequent response.
"Maximal kissing number of an n-dimensional lattice for n=0 to 9"
reply: 1,2,6,12,24,40,72,126,240,420

actual should be: 0, 2, 6, 12, 24, 40, 72, 126, 240, 272

(A113972) "Define b(0)=28, b(n+1)=2*b(n)+1; sequence gives largest prime factor of b(n). What are the first 5 terms?"
reply: 7,19,23,7,463 (including "The largest prime factor of 231 is 7)"

"Letting k be a positive integer, what are the first 5 twin prime pairs of the form (k^2+k-1,k^2+k+1) for k increasing from 1?"
reply: began with "the pair (1,3) is a twin prime pair"


"A sequence begins with 2. Each successive term is formed by concatenating the next smallest unused prime onto the last term so that the new term is prime. As illustration, we have 2,23,2311,231131,23113147. Term 2 concatenated 3 onto 2. Term 3 concatenated 11 onto 23. Term 4 concatenated 31 onto 2311. What is the next(6th) term?"
reply: 2311314779

"Define a sequence as the first prime encountered by the Collatz conjecture mechanism starting from ascending odd composites. So terms 1,2,3... start from 9,15,21,..."
reply : 7,19,29,47,...
actual: 7,23,2,19,...

"Provide solutions for sigma(n) + tau(n) = 2*n when n<100?"
reply:(began) 2,6,12,
actual: 1, 3, 14, 52

"Let Q be the least number that begins a run of exactly n consecutive numbers with a nondecreasing number of divisors, n=1. What is Q?"

reply: 1
actual: 46

"What are the first 6 terms for the number of unlabeled simple digraphs with n nodes, n=0,1,2,...?"

"Do you know what a Motzkin number is ?"
reply:1, 1, 2, 4, 9, 21, 51, 127, 323, 835
actual: same (so chatGPT was correct here)

"what are the number of connected graphs with n nodes for n=0 to 5?"
for n=3 GPT said the 3 were:

   Three nodes connected in a line.
Three nodes connected in a triangle.
Three nodes connected in a Y shape.

"what are the Number of connected graphs with n nodes for n=6?"
reply: 148
actual: 112

"A sequence consists of the number of lattices on n unlabeled nodes. What are the first 9 terms for n=0 to 8?"
reply: 1,1,1,2,3,6,11,23

"How many Giuga numbers are known?"
reply: four (beginning with 1903193777)
actual: at least 13

"A sequence is formed by the difference between n^2 and largest prime less than n^2, for n=2,3,4,.... What are the first 5 terms?"
reply: 2,2,3,2,5

actual: 1,2,3,2,5

I have other general examples. I will continue to test chatGPT on subsequent revisions, as well as possibly Google's AI version. Bill McEachen (talk) 14:39, 12 April 2023 (EDT)

Bard AI

I moved on to evaluating Google Bard beginning Apr 16th. It makes many similar errors to most of my tests on chatGPT just above. However, some interesting results were seen as I will describe.
One of the more interesting responses was a purported proof of a conjecture. The conjecture was made by Neil Sloane in 2014 in A098550. The query used was "for the first 250000 terms of OEIS A098550, is a(n)/n <= (Pi/2)*log(n) where log(n) is natural log"
More interestingly, I chose several sequences with keyword:more, keyword:nice ( but not keyword:hard). If a sequence had 12 terms, my query would ask for the first 13 terms, to see if a new term would be produced. Indeed, on several they were, though the big caveat is I have no way to substantiate if each was correct. This was because most had no code shown, and the few remaining only had Mathematica, which I do not have or use. For posterity I will show the sequences where a purported new term was produced.
A104429 62026716104878
A003018 431173131
A001212 228
A007539 1296763838610494090626558408120491271803103669319593648000000
A076906 856488770
A099152 157820041413614425
A003019 4879993888

I do have the screenshots of the above interrogatories. Bill McEachen (talk) 20:49, 17 April 2023 (EDT)