A226863 -id:A226863 - cp's OEIS frontend

A113502 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).

3, 6, 9, 10, 12, 15, 18, 20, 21, 24, 27, 28, 30, 33, 36, 39, 40, 42, 45, 48, 50, 51, 54, 55, 56, 57, 60, 63, 66, 69, 70, 72, 75, 78, 80, 81, 84, 87, 90, 91, 93, 96, 99, 100, 102, 105, 108, 110, 111, 112, 114, 117, 120, 123, 126, 129, 130, 132, 135, 136, 138, 140, 141

Offset: 1

Leroy Quet, Jan 10 2006

nonn

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

Multiples of A226863. - Charles R Greathouse IV, Jul 29 2016

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

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A112886, A000217, A013929, A226863.

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

is(n)=fordiv(n,d, if(ispolygonal(d,3) && d>1, return(1))); 0 \\ Charles R Greathouse IV, Jul 29 2016

a(n) = A088723(n)/2. - Ray Chandler, May 29 2008

More terms from Farideh Firoozbakht, Jan 12 2006

A137281 Numbers k such that T(k) is not divisible by T(i), 1 < i < k, where T(k) = k-th triangular number A000217(k).

Original entry on oeis.org

2, 4, 7, 10, 13, 16, 22, 25, 28, 31, 34, 37, 43, 46, 49, 52, 58, 61, 67, 70, 73, 76, 82, 85, 88, 94, 97, 103, 106, 118, 121, 127, 130, 133, 136, 142, 145, 148, 151, 157, 163, 166, 169, 172, 178, 187, 190, 193, 196, 202, 205, 208, 211, 214, 217, 226, 229, 232, 238, 241

Offset: 1

Zak Seidov, Mar 14 2008

nonn

All terms > 5 in A005383 are here. - Zak Seidov, Jun 20 2013
All terms except 2 are congruent to 1 (mod 3). This is required for 3 not to be a divisor of T(n). - Franklin T. Adams-Watters, Dec 10 2019
Conjecture: a(n) ~ C * n * log(n) for some constant C, in analogy with the prime number theorem (see A000040). - Harry Richman, Mar 05 2025

			T(4)=10 is not divisible by lesser T's 3, 6;
T(7)=28 is not divisible by lesser T's 3, 6, 10, 15, 21.

Zak Seidov, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Cf. A000217, A005383, A076982, A226863.

nn = 241; tri = Table[n*(n+1)/2, {n, nn}]; Select[Range[2, nn], ! MemberQ[Mod[tri[[#]], Take[tri, {2, # - 1}]], 0] &] (* T. D. Noe, Apr 12 2011 *)

n such that A076982(n) = 2. - T. D. Noe, Apr 12 2011

A000217(a(n)) = A226863(n). - Zak Seidov, Jun 20 2013

cp's OEIS Frontend

A113502 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).

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