A selection of procedurally generated conjectures

While sequencedb.net consists of procedurally generated conjectures for more than 150000 entries in the Online Encyclopedia of Integer Sequences (OEIS), their quality vary and new interesting formulas can be hard to spot.

To narrow down the search I have added a few keywords to help. For example

!oeis-mention-formula oeis-keywords:!formulas min-matching:200 op-oeis

reveals conjectures for more than 5000 24000 OEIS entries that have no formula field (yet) neither mentions any of the conjectured A-numbers. Also, each conjecture matches the corresponding OEIS entry with at least 200 terms.

Some conjectures are obvious or wrong, but the list also contains some neat constructions!

The purpose of this post is the get help to identify relevant conjectures and getting them into the OEIS.

Example 1 multiples of 123456789

To give an example from the list

A053654 Multiples of 123456789.

123456789, 246913578, 370370367, 493827156, 617283945, 740740734, 864197523, 987654312, 1111111101...


The conjecture suggests that the multiples of 123456789 can be described from the sequence of zeroless pandigital numbers combined with the highest power of twos dividing even numbers.

If you think it looks interesting see steps below on how you can get it into the OEIS and get your name associated to the formula ;) Or pick another one of the 5322 conjectures!

Example 2 the Collatz conjecture

Perhaps you don’t care about some multiples of 123456789 and want to tackle something… grander. Then you can scroll through the entries until you see something interesting or if you know what you are looking for you can refine the search.

Let’s say you are interested in the Collatz conjecture, then you can just append the query above with ‘Collatz’

!oeis-mention-formula oeis-keywords:!formulas min-matching:200 op-oeis collatz

Then you will get around 50 Collatz related entries without a formula in OEIS for example

A273216 Initial prime numbers encountered by Collatz (3x+1) evaluation at odd, composite, ascending starts.

7, 23, 2, 19, 41, 19, 53, 59, 17, 37, 29, 83, 43, 137, 37, 13, 113, 29, 61, 2, 131...


You can tweak the queries plenty, for example here are about 200 OEIS entries that mentions Collatz in their title, each with up to 5 conjectures

collatz min-ops-1

See here for a lot of ways to search. This way it’s pretty fun to explore different problems.

Submission to OEIS

The steps I usually take for OEIS submissions are:

  1. Does the conjecture make any sense, does it add value?
  2. Check through the OEIS entry to make sure it’s not already there (check title, comments, formula, programs, cross-refs etc).
  3. Implement the conjectured formula in a language that supports arbitrary precision.
  4. Verify the conjecture with as many terms as realistically possible.
  5. Simplify the formula, rewrite it in terms of other A-numbers etc. Make sure the offsets are correct (see offset field in the OEIS entries, all conjectured A-numbers are 0-offsetted). Verify the rewritten formula.
  6. Try to prove the conjecture (this is usually the most time consuming for me).
  7. Submit the formula to the OEIS after carefully reading their style sheet.