cp's OEIS Frontend

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

%I A333655 #22 Jun 02 2025 15:23:14
%S A333655 1,4,24,36,48,180,240,720,840,1260,1680,7560,10080,15120,20160,25200,
%T A333655 27720,45360,50400,83160,110880,166320,221760,277200,332640,498960,
%U A333655 554400,665280,1081080,2162160,2882880,3603600,6486480,7207200,8648640,10810800,14414400
%N A333655 Highly composite numbers (A002182) that are not superior highly composite numbers (A002201).
%C A333655 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.
%C A333655 This sequence is the same as A189228 until n=12, for which a(12) = 7560 and A189228(12) = 10080.
%H A333655 Iain Fox, <a href="/A333655/b333655.txt">Table of n, a(n) for n = 1..9780</a> (calculated using b-files of A002182 and A002201)
%H A333655 David Terr, <a href="https://mathworld.wolfram.com/SuperiorHighlyCompositeNumber.html">Superior Highly Composite Number</a>.
%H A333655 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HighlyCompositeNumber.html">Highly Composite Number</a>.
%H A333655 Wikipedia, <a href="https://en.wikipedia.org/wiki/Highly_composite_number">Highly composite number</a>.
%H A333655 Wikipedia, <a href="https://en.wikipedia.org/wiki/Superior_highly_composite_number">Superior highly composite number</a>.
%e A333655 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.
%o A333655 (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)
%Y A333655 Cf. A000005, A189228.
%Y A333655 Highly composite numbers: A002182.
%Y A333655 Superior highly composite numbers: A002201.
%K A333655 nonn
%O A333655 1,2
%A A333655 _Iain Fox_, Aug 23 2020