A076711 Highly composite triangular numbers: triangular numbers where the number of divisors increases to a record.

1, 3, 6, 28, 36, 120, 300, 528, 630, 2016, 3240, 5460, 25200, 73920, 157080, 437580, 749700, 1385280, 1493856, 2031120, 2162160, 17907120, 76576500, 103672800, 236215980, 842161320, 3090906000, 4819214400, 7589181600, 7966312200, 13674528000, 20366564400

Offset: 1

Author

Shyam Sunder Gupta, Oct 26 2002

Keywords

• nonn

Examples

			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.
		

Links

• Charles R Greathouse IV and Donovan Johnson, Table of n, a(n) for n = 1..70 (terms < 10^23; first 55 terms from _Charles R Greathouse IV_).
• Mark Dominus, More sawed-off shotguns, 2007.
• David Eppstein, Triangular numbers with many factors, 2007.
• Shyam Sunder Gupta Fascinating Triangular Numbers.

Crossrefs

Cf. A101755, A101756.

Programs

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

Extensions

Name clarified by Andrew Howroyd, Sep 20 2024