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
Michael De Vlieger, Table of n, a(n) for n = 1..16384 (first 137 terms from Giovanni Resta).
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
KEYWORD
nonn
AUTHOR
Pham G. Hoang, Jun 02 2019
EXTENSIONS
a(40)-a(43) from Giovanni Resta, Jun 04 2019
STATUS
approved