A291834 First positions of records in A252665.

1, 4, 8, 12, 16, 24, 36, 48, 72, 96, 120, 144, 216, 240, 288, 360, 432, 480, 576, 720, 1080, 1440, 2160, 2520, 2880, 3600, 4320, 5040, 7200, 7560, 8640, 10080, 14400, 15120, 20160, 25200, 30240, 40320, 50400, 60480, 75600, 80640, 90720, 100800, 120960, 151200, 181440

Offset: 1

Author

Michael De Vlieger, Sep 03 2017

Keywords

• nonn

Comments

Distinct from A033833; first term not in A033833 is a(24) = 2520. There appear to be increasingly many terms a(n) not in A033833 as n increases.

The terms 2520, 7560, 25200, 221760, 665280, 8648640, ... are not in A033833 but are in A002182. The term 3600 is the smallest that is in neither A033833 nor A002182, but in A007416. The term 831600 is the smallest that is in none of the three aforementioned sequences.

Conjectures based on a(n) < 10^7:

1. Numbers in a(n) are products of the first several consecutive primes p.

2. Outside of a(1), the least prime factor of a(n) has multiplicity > 1. This implies no primes, primorials, or squarefree a(n) for n > 1.

3. The greatest prime factor of a(n) generally has multiplicity 1. Note, however, exceptions in a(n) for n = {1, 2, 3, 5, 7, 9, 12, 13, 15, 17, 19, 26, 29, 33, 73, ...}.

4. The multiplicities of prime factors p of m generally decrease or stay the same as p increases.

See "Records and first positions of records in A252665" for more information. - Michael De Vlieger, Sep 03 2017

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..76

Crossrefs

Cf. A002182, A007416, A033833, A252665, A291833.

Programs

Mathematica

With[{s = Array[f[#, #, 5] &, 10^4]}, Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]]