A116990 Indices of triangular numbers whose sum of divisors is square.

1, 2, 11, 20, 40, 68, 92, 212, 236, 253, 266, 321, 328, 452, 582, 589, 596, 668, 695, 716, 782, 788, 836, 928, 932, 970, 991, 1012, 1065, 1076, 1173, 1264, 1300, 1336, 1388, 1436, 1490, 1549, 1796, 1854, 1927, 1995, 2159, 2228, 2252, 2468, 2545, 2588

Offset: 1

Author

Jonathan Vos Post, Apr 04 2006

Keywords

• easy
• nonn
• less

Comments

From Zak Seidov, Oct 19 2010: (Start)

A074285(n) = A000203(A000217(n)) = s^2.

Corresponding values of s begin: 1,2,12,24,42,72,96,216,240,192,240,288,336,456,504, 480,600,672,840,720,720,792,960,930,936,756,992,936,1008,1080,... (are most values of s multiples of 3?).

(End)

Examples

			a(1) = 1 because sigma(1*2/2) = sigma(1) = 1 = 1^2,
a(2) = 2 because sigma(2*3/2) = sigma(3) = 2^2,
a(3) = 11 because sigma(11*12/2) = sigma(66) = 144 = 12^2.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..561 from Zak Seidov)

Crossrefs

See also: A000217 Triangular numbers: a(n) = C(n+1,2) = n(n+1)/2 = 0+1+2+...+n. A074285 Sum of the divisors of n-th triangular number. A083675 Triangular number for which the sum of the proper divisors is also a triangular number. A000203 sigma(n) = sum of divisors of n. Also called sigma_1(n).

Cf. A000203, A000217, A000290, A074285, A083675.

Programs

Maple

with(numtheory): a:=proc(n) if type(sqrt(sigma(n*(n+1)/2)),integer)=true then n else fi end: seq(a(n),n=0..3100); # Emeric Deutsch, Apr 06 2006

Mathematica

Flatten@ Position[Accumulate[Range@ 2600], n_ /; IntegerQ@ Sqrt@ DivisorSigma[1, n] == True] (* Michael De Vlieger, Mar 17 2015 *)
Select[Range[2600],IntegerQ[Sqrt[DivisorSigma[1,(#(#+1))/2]]]&] (* Harvey P. Dale, Nov 19 2022 *)

PARI

for(n=1,1000,if(issquare(sigma(n*(n+1)/2)),print1(n","))) \\ Zak Seidov, Mar 21 2015

Formula

n such that A074285(n) is in A000290.

n such that sum( d | A000217(n), d ) is in A000290.

n such that A000203(A000217(n)) is in A000290.

n such that sum( d | n*(n+1)/2, d ) = k^2 for integer k.

Extensions

More terms from Emeric Deutsch, Apr 06 2006

Incorrect term 0 removed by Michel Marcus, Mar 17 2015