cp's OEIS Frontend

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

%I A116990 #38 Nov 19 2022 18:08:08
%S A116990 1,2,11,20,40,68,92,212,236,253,266,321,328,452,582,589,596,668,695,
%T A116990 716,782,788,836,928,932,970,991,1012,1065,1076,1173,1264,1300,1336,
%U A116990 1388,1436,1490,1549,1796,1854,1927,1995,2159,2228,2252,2468,2545,2588
%N A116990 Indices of triangular numbers whose sum of divisors is square.
%C A116990 From _Zak Seidov_, Oct 19 2010: (Start)
%C A116990 A074285(n) = A000203(A000217(n)) = s^2.
%C A116990 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?).
%C A116990 (End)
%H A116990 Amiram Eldar, <a href="/A116990/b116990.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..561 from Zak Seidov)
%F A116990 n such that A074285(n) is in A000290.
%F A116990 n such that sum( d | A000217(n), d ) is in A000290.
%F A116990 n such that A000203(A000217(n)) is in A000290.
%F A116990 n such that sum( d | n*(n+1)/2, d ) = k^2 for integer k.
%e A116990 a(1) = 1 because sigma(1*2/2) = sigma(1) = 1 = 1^2,
%e A116990 a(2) = 2 because sigma(2*3/2) = sigma(3) = 2^2,
%e A116990 a(3) = 11 because sigma(11*12/2) = sigma(66) = 144 = 12^2.
%p A116990 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
%t A116990 Flatten@ Position[Accumulate[Range@ 2600], n_ /; IntegerQ@ Sqrt@ DivisorSigma[1, n] == True] (* _Michael De Vlieger_, Mar 17 2015 *)
%t A116990 Select[Range[2600],IntegerQ[Sqrt[DivisorSigma[1,(#(#+1))/2]]]&] (* _Harvey P. Dale_, Nov 19 2022 *)
%o A116990 (PARI) for(n=1,1000,if(issquare(sigma(n*(n+1)/2)),print1(n","))) \\ _Zak Seidov_, Mar 21 2015
%Y A116990 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).
%Y A116990 Cf. A000203, A000217, A000290, A074285, A083675.
%K A116990 easy,nonn,less
%O A116990 1,2
%A A116990 _Jonathan Vos Post_, Apr 04 2006
%E A116990 More terms from _Emeric Deutsch_, Apr 06 2006
%E A116990 Incorrect term 0 removed by _Michel Marcus_, Mar 17 2015