A101755 -id:A101755 - cp's OEIS frontend

A063440 Number of divisors of n-th triangular number.

1, 2, 4, 4, 4, 4, 6, 9, 6, 4, 8, 8, 4, 8, 16, 8, 6, 6, 8, 16, 8, 4, 12, 18, 6, 8, 16, 8, 8, 8, 10, 20, 8, 8, 24, 12, 4, 8, 24, 12, 8, 8, 8, 24, 12, 4, 16, 24, 9, 12, 16, 8, 8, 16, 24, 24, 8, 4, 16, 16, 4, 12, 36, 24, 16, 8, 8, 16, 16, 8, 18, 18, 4, 12, 24, 16, 16, 8, 16, 40, 10, 4, 16

Offset: 1

Henry Bottomley, Jul 24 2001

a(n) = 4 iff either n is in A005383 or n/2 is in A005384.
a(n) is odd iff n is in A001108.
a(n) = 6 if either n = 18 or n = q^2 where q is in A048161 or n = 2 q^2 - 1 where q is in A106483. - Robert Israel, Oct 26 2015
From Bernard Schott, Aug 29 2020: (Start)
a(n-1) is the number of solutions in positive integers (x, y, z) to the simultaneous equations (x + y - z = n, x^2 + y^2 - z^2 = n) for n > 1. See the British Mathematical Olympiad link. In this case, one always has z > x and z > y.
For n = 12 as in the Olympiad problem, the a(11) = 8 solutions are (13,78,79), (14,45,47), (15,34,37), (18,23,29), (23,18,29), (34,15,37), (45,14,47), (78,13,79). (End)

			a(6) = 4 since 1+2+3+4+5+6 = 21 has four divisors {1,3,7,21}.

Steve Dinh, The Hard Mathematical Olympiad Problems And Their Solutions, AuthorHouse, 2011, Problem 2 of the British Mathematical Olympiad 2007, page 28.

Ray Chandler, Table of n, a(n) for n = 1..10000
British Mathematical Olympiad 2007/2008, Round 1, Problem 2.
Index to sequences related to Olympiads.

Cf. A000005, A000217.

Cf. A001108, A005383, A005384, A048161, A060778, A081978 (greedy inverse), A106483, A101755 (indices of records), A101756 (records).

seq(numtheory:-tau(n*(n+1)/2), n=1..100); # Robert Israel, Oct 26 2015

DivisorSigma[0,#]&/@Accumulate[Range[90]] (* Harvey P. Dale, Apr 15 2019 *)

for (n=1, 10000, write("b063440.txt", n, " ", numdiv(n*(n + 1)/2)) ) \\ Harry J. Smith, Aug 21 2009

a(n)=factorback(apply(numdiv,if(n%2,[n,(n+1)/2],[n/2,n+1]))) \\ Charles R Greathouse IV, Dec 27 2014

vector(100, n, numdiv(n*(n+1)/2)) \\ Altug Alkan, Oct 26 2015

a(n) = A000005(A000217(n)).
From Robert Israel, Oct 26 2015: (Start)
a(2k) = A000005(k)*A000005(2k+1).
a(2k+1) = A000005(2k+1)*A000005(k+1).
gcd(a(2k), a(2k+1)) = A000005(2k+1) * A060778(k). (End)

A101756 Number of divisors of highly composite triangular numbers.

Original entry on oeis.org

1, 2, 4, 6, 9, 16, 18, 20, 24, 36, 40, 48, 90, 112, 128, 144, 162, 168, 192, 240, 320, 480, 576, 648, 768, 1024, 1280, 1344, 1512, 1536, 1728, 1920, 2304, 2880, 3072, 3200, 3456, 3888, 4480, 5760, 6144, 6912, 7680, 8640, 9216, 13824, 16128, 23040, 25088

Offset: 1

Ray Chandler, Dec 22 2004

nonn

Charles R Greathouse IV and Donovan Johnson, Table of n, a(n) for n = 1..70 (first 55 terms from _Charles R Greathouse IV_).

Cf. A076711, A101755.

t1=1;r=0;for(n=1,2e9,t2=n+1;t2=numdiv(t2/if(n%2,2,1));if(t1*t2>r,r=t1*t2;print1(r", "));t1=t2) \\ Charles R Greathouse IV, Feb 01 2013

a(n) = A000005(A076711(n)) = A000005(A000217(A101755(n))).

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

Original entry on oeis.org

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

Shyam Sunder Gupta, Oct 26 2002

nonn

			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.

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.

Cf. A101755, A101756.

{ 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

Name clarified by Andrew Howroyd, Sep 20 2024

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A063440 Number of divisors of n-th triangular number.

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A101756 Number of divisors of highly composite triangular numbers.

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

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