A113502 - OEIS

%I #27 Dec 13 2017 09:02:29

%S 3,6,9,10,12,15,18,20,21,24,27,28,30,33,36,39,40,42,45,48,50,51,54,55,

%T 56,57,60,63,66,69,70,72,75,78,80,81,84,87,90,91,93,96,99,100,102,105,

%U 108,110,111,112,114,117,120,123,126,129,130,132,135,136,138,140,141

%N A number n is included if at least one of its divisors > 1 is a triangular number (i.e., is of the form m(m+1)/2, m >= 2).

%C A number n is in the sequence iff it is not a "triangle-free" positive integer.

%C Multiples of A226863. - _Charles R Greathouse IV_, Jul 29 2016

%H Charles R Greathouse IV, <a href="/A113502/b113502.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A088723(n)/2. - _Ray Chandler_, May 29 2008

%e 12 is included because its divisors are 1, 2, 3, 4, 6 and 12, two of which (3 and 6) are triangular numbers > 1.

%t v={};Do[If[b=Select[Divisors[n], #>1 && IntegerQ[(1+8#)^(1/2)]&]; b!={}, AppendTo[v, n]], {n, 200}]; v (* _Farideh Firoozbakht_, Jan 12 2006 *)

%t Select[Range[200],AnyTrue[Rest[Divisors[#]],OddQ[Sqrt[8#+1]]&]&] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Dec 13 2017 *)

%o (PARI) is(n)=fordiv(n,d, if(ispolygonal(d,3) && d>1, return(1))); 0 \\ _Charles R Greathouse IV_, Jul 29 2016

%Y Cf. A112886, A000217, A013929, A226863.

%K nonn

%O 1,1

%A _Leroy Quet_, Jan 10 2006

%E More terms from _Farideh Firoozbakht_, Jan 12 2006