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

1,2

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)

Links

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

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.

Example

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.

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 *)

Prog

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

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.

Sequence in context: A279774 A092595 A175927 * A327264 A279772 A226416

Adjacent sequences: A116987 A116988 A116989 * A116991 A116992 A116993

Keyword

easy,nonn,less

Author

Jonathan Vos Post, Apr 04 2006

Extensions

More terms from Emeric Deutsch, Apr 06 2006

Incorrect term 0 removed by Michel Marcus, Mar 17 2015

Status

approved