A116990 - OEIS

%I #38 Nov 19 2022 18:08:08

%S 1,2,11,20,40,68,92,212,236,253,266,321,328,452,582,589,596,668,695,

%T 716,782,788,836,928,932,970,991,1012,1065,1076,1173,1264,1300,1336,

%U 1388,1436,1490,1549,1796,1854,1927,1995,2159,2228,2252,2468,2545,2588

%N Indices of triangular numbers whose sum of divisors is square.

%C From _Zak Seidov_, Oct 19 2010: (Start)

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

%C 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 (End)

%H Amiram Eldar, <a href="/A116990/b116990.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..561 from Zak Seidov)

%F n such that A074285(n) is in A000290.

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

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

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

%e a(1) = 1 because sigma(1*2/2) = sigma(1) = 1 = 1^2,

%e a(2) = 2 because sigma(2*3/2) = sigma(3) = 2^2,

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

%p 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 Flatten@ Position[Accumulate[Range@ 2600], n_ /; IntegerQ@ Sqrt@ DivisorSigma[1, n] == True] (* _Michael De Vlieger_, Mar 17 2015 *)

%t Select[Range[2600],IntegerQ[Sqrt[DivisorSigma[1,(#(#+1))/2]]]&] (* _Harvey P. Dale_, Nov 19 2022 *)

%o (PARI) for(n=1,1000,if(issquare(sigma(n*(n+1)/2)),print1(n","))) \\ _Zak Seidov_, Mar 21 2015

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

%K easy,nonn,less

%O 1,2

%A _Jonathan Vos Post_, Apr 04 2006

%E More terms from _Emeric Deutsch_, Apr 06 2006

%E Incorrect term 0 removed by _Michel Marcus_, Mar 17 2015