A308508 Numbers (including primorials) that satisfy "three rules that highly composite numbers must have" (see Comments below) but are not highly composite numbers.

30, 96, 192, 210, 384, 420, 480, 768, 960, 1080, 1440, 1536, 1920, 2160, 2310, 2880, 3072, 3360, 3840, 4320, 4620, 5760, 6144, 6300, 6480, 6720, 7680, 8640, 9240, 11520, 12288, 12600, 12960, 13440, 13860, 15360, 17280, 18480, 23040, 24576, 25920, 26880, 30030

Offset: 1

Author

Pham G. Hoang, Jun 02 2019

Keywords

• nonn

Comments

The three rules that the highly composite numbers (and this sequence too) must have are:

- The primes in the product have to be consecutive and start with 2,

- The exponents of the primes must be decreasing,

- The final exponent of the primes must be 1 (except 4 = 2^2 and 36 = 2^2 * 3^2, but those are highly composite numbers and are excluded).

Except for numbers of the form 2^n * 3, the terms are divisible by 10, because a(n) has the form of 2^n * 3^m * 5^j * c = (2 * 5) * 2^(n - 1) * 3^m * 5^(j - 1) * c.

Except for terms that are primorials, all others are divisible by 12, because a(n) has the form 2^n * 3^m * c = (2^2 * 3) * 2^{n - 2} * 3^(m - 1) * c.

All terms are divisible by 6, because a(n) has the form 2^n * 3^m * c = (2 * 3) * 2^(n - 1) * 3^(m - 1) * c.

It seems that eulerphi(a(n)) is always divisible by 8.

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..16384 (first 137 terms from Giovanni Resta).

Crossrefs

Contains A002110(n) (primorials).

Does not contain A002182(n) (highly composite numbers).

Programs

Mathematica

limit = 2*10^5; hc = {1}; r=1; Do[t = DivisorSigma[0, n]; If[t > r, r=t; AppendTo[hc, n]], {n, 2, limit, 2}]; ok[n_] := Block[{f = FactorInteger[n]}, ! MemberQ[hc, n] && f[[-1, 2]] == 1 && Max[ Differences[Last /@ f]] <= 0 && Union[ Differences[ PrimePi[ First /@ f]]] == {1}]; Select[Range[2, limit, 2], ok] (* Giovanni Resta, Jun 10 2019 *)

Extensions

a(40)-a(43) from Giovanni Resta, Jun 04 2019