Primorial

2018-05-30 18:15

tl;dr: This morning I tweeted:

What occasioned this tweet was exploring a mild curiosity about how a sequence of products of the primes would grow. In my own head I had provisionally dubbed this sequence “Prime Factorial,” because it is the product of all numbers up to a given point, but in this case using as input the primes instead of the natural numbers.

Spoiler: It grows damn fast.

Then I looked for that sequence in the Online Encyclopedia of Integer Sequences, and surprise surprise: It is well known.

Those numbers are called Primorials (A002110), apparently in analogy with factorials, as I had independently invented. n primorial is written n#, where n is the nth prime.

Pro tip: anything you wonder in an idle moment concerning the primes has been explored with surpassing thoroughness before you were born. Ramanujan spent much of his short life obsessed with the primes, and ditto Erdős his long life. And let’s not speak of Reimann, and probably 10,000 others living and departed.

Side trips from the OEIS included the below hypnotic 8.5-min. movie. (The soundtrack is a rendering in audio of Recaman’s sequence, A005132.)


The movie was produced in 2010 to celebrate the launch of the revamped website under the aegis of the OEIS Foundation, which took the project over from its founder Neil J.A. Sloane. Sloane had started collecting integer sequences in 1964 while at Cornell. In 1973 he authored the Handbook of Integer Sequences, which characterized 2,372 of them. Today the OEIS has entries for 304,836. And it is searchable.

OEIS.org enjoys over 8,000 registered users, who presumably signed up because they want to comment or edit or suggest a new sequence for the database. Something like 20,000 new entries per month are added, after vetting by a volunteer staff of 100-odd editors. (Yes, I think we can safely assume they are odd to a man/woman/*.)

  

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