%I #22 Sep 20 2024 14:15:11

%S 1,3,6,28,36,120,300,528,630,2016,3240,5460,25200,73920,157080,437580,

%T 749700,1385280,1493856,2031120,2162160,17907120,76576500,103672800,

%U 236215980,842161320,3090906000,4819214400,7589181600,7966312200,13674528000,20366564400

%N Highly composite triangular numbers: triangular numbers where the number of divisors increases to a record.

%H Charles R Greathouse IV and Donovan Johnson, <a href="/A076711/b076711.txt">Table of n, a(n) for n = 1..70</a> (terms < 10^23; first 55 terms from _Charles R Greathouse IV_).

%H Mark Dominus, <a href="http://blog.plover.com/oops/triangular-phi.html">More sawed-off shotguns</a>, 2007.

%H David Eppstein, <a href="https://11011110.github.io/blog/2007/04/05/triangular-numbers-with.html">Triangular numbers with many factors</a>, 2007.

%H Shyam Sunder Gupta <a href="http://www.shyamsundergupta.com/triangle.htm">Fascinating Triangular Numbers</a>.

%e a(4)=28: 28 is a triangular number and has 6 divisors. Number of divisors of all triangular numbers less than 28 is less than 6. So 28 is a highly composite triangular number.

%o (PARI) { my(r=0); for(n=1, 1e7, my(t=if(n%2, numdiv(n)*numdiv((n+1)/2), numdiv(n/2)*numdiv(n+1))); if(t>r, r=t; print1(n*(n+1)/2", "))) } \\ _Charles R Greathouse IV_, Feb 01 2013

%Y Cf. A101755, A101756.

%K nonn

%O 1,2

%A _Shyam Sunder Gupta_, Oct 26 2002

%E Name clarified by _Andrew Howroyd_, Sep 20 2024