|
|
A308508
|
|
Numbers (including primorials) that satisfy "three rules that highly composite numbers must have" (see Comments below) but are not highly composite numbers.
|
|
1
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
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
|
|
|
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 *)
|
|
CROSSREFS
|
Does not contain A002182(n) (highly composite numbers).
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|