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A161894 Small factors of some highly composite numbers.

%I A161894 #15 Aug 21 2019 05:48:30
%S A161894 1,1,2,4,12,24,72,72,288,1440,1440,10080,10080,10080,30240,30240,
%T A161894 100800,100800,907200,907200,907200,6350400,9979200,9979200,69854400,
%U A161894 69854400,69854400,129729600,129729600,259459200,1210809600,1816214400,3632428800,3632428800
%N A161894 Small factors of some highly composite numbers.
%C A161894 Definition: Let p(n) be the product of first n primes (primorial A002110) then c(n)=a(n)*p(n) is the unique number such that every other number smaller than c(n) has fewer divisors and all c(k) with k < n have fewer distinct factors than c(n). (tau(n) and littleomega(c(n)) increase simultaneously to a new record.)
%H A161894 Amiram Eldar, <a href="/A161894/b161894.txt">Table of n, a(n) for n = 1..1229</a>
%e A161894 For example a(24)=9979200,
%e A161894 p(24) = 23768741896345550770650537601358310 and
%e A161894 c(24) = 237193029132011520250475844831474847152000.
%e A161894 Every other number n < c(24) has fewer than tau(c(24))=905969664
%e A161894 divisors and c(1),...,c(23) have fewer than 24 distinct factors.
%e A161894 The sequence of corresponding highly composite numbers starts
%e A161894 2
%e A161894 6
%e A161894 60
%e A161894 840
%e A161894 27720
%e A161894 720720
%e A161894 36756720
%e A161894 698377680
%e A161894 64250746560
%e A161894 9316358251200
%e A161894 288807105787200
%e A161894 74801040398884800
%e A161894 3066842656354276800
%e A161894 131874234223233902400
%e A161894 18594267025475980238400
%e A161894 985496152350226952635200
%e A161894 193814243295544634018256000
%e A161894 11822668841028222675113616000
%e A161894 7129069311140018273093510448000
%e A161894 506163921090941297389639241808000
%e A161894 36949966239638714709443664651984000
%e A161894 20433331330520209234322346552547152000
%e A161894 2665090214966421575848043200353649968000
%e A161894 237193029132011520250475844831474847152000
%e A161894 For n > 1 this sequence is conjectured to be a subsequence of A161812.
%Y A161894 Cf. A002182, A161812.
%K A161894 easy,nonn
%O A161894 1,3
%A A161894 _Peter Luschny_, Jun 21 2009
%E A161894 More terms from _Amiram Eldar_, Aug 21 2019