A333655 Highly composite numbers (A002182) that are not superior highly composite numbers (A002201).

1, 4, 24, 36, 48, 180, 240, 720, 840, 1260, 1680, 7560, 10080, 15120, 20160, 25200, 27720, 45360, 50400, 83160, 110880, 166320, 221760, 277200, 332640, 498960, 554400, 665280, 1081080, 2162160, 2882880, 3603600, 6486480, 7207200, 8648640, 10810800, 14414400

Offset: 1

Author

Iain Fox, Aug 23 2020

Keywords

• nonn

Comments

For a number n to be in this sequence, it must have the following conditions be true, where d(n) represents the number of divisors of n (A000005): d(n) > d(k), for all k < n, and there does not exist a number e > 0 such that d(n)/n^e >= d(k)/k^e for k < n and d(n)/n^e > d(k)/k^e for k > n.

This sequence is the same as A189228 until n=12, for which a(12) = 7560 and A189228(12) = 10080.

Examples

			4 is in the sequence because it has three factors, more than any preceding number, making it highly composite, but it is not a superior highly composite number.
		

Links

• Iain Fox, Table of n, a(n) for n = 1..9780 (calculated using b-files of A002182 and A002201)
• David Terr, Superior Highly Composite Number.
• Eric Weisstein's World of Mathematics, Highly Composite Number.
• Wikipedia, Highly composite number.
• Wikipedia, Superior highly composite number.

Crossrefs

Cf. A000005, A189228.

Highly composite numbers: A002182.

Superior highly composite numbers: A002201.

Programs

PARI

lista(nn)=my(v, w=[1,2,4], r=1, p=primes(primepi(2^log(nn)))); v=setminus(Set(vector(nn, i, prod(n=1, primepi(2^log(i)), p[n]^floor(1/(p[n]^(1/log(i))-1))))), [1]); forstep(x=6, v[#v], 6, if(numdiv(x)>r, r=numdiv(x); w=setunion(w, [x]))); setminus(w, v)