A112886 Positive integers that have no triangular divisors > 1.

1, 2, 4, 5, 7, 8, 11, 13, 14, 16, 17, 19, 22, 23, 25, 26, 29, 31, 32, 34, 35, 37, 38, 41, 43, 44, 46, 47, 49, 52, 53, 58, 59, 61, 62, 64, 65, 67, 68, 71, 73, 74, 76, 77, 79, 82, 83, 85, 86, 88, 89, 92, 94, 95, 97, 98, 101, 103, 104, 106, 107, 109, 113, 115, 116, 118, 119

Offset: 1

Author

Leroy Quet, Jan 10 2006

Keywords

• nonn

Comments

Sequence consists of those positive integers not in A113502.

(Pi^(3/2))/density is empirically close to 10. - Richard Peterson, Apr 06 2025

Examples

			14 is included because the divisors of 14 are 1, 2, 7 and 14, none of which are triangular numbers > 1.
		

Links

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

Crossrefs

Cf. A113502, A000217, A005117, A132895.

Programs

Mathematica

v={};Do[If[b=Select[Divisors[n], #>1 && IntegerQ[(1+8#)^(1/2)]&]; b=={}, AppendTo[v, n]], {n, 138}]; v (Firoozbakht)
tfpnQ[n_]:=Module[{nn=120,trnos},trnos=Rest[Accumulate[ Range[ (Sqrt[8nn+1]-1)/2+1]]]; Intersection[ Divisors[ n],trnos]=={}]; Select[Range[ 120], tfpnQ] (* Harvey P. Dale, Jul 17 2015 *)

PARI

is(n)=fordiv(n, d, if(ispolygonal(d, 3) && d>1, return(0))); 1 \\ Charles R Greathouse IV, Jul 31 2016

Python

from itertools import count, islice
from sympy import divisors
from sympy.ntheory.primetest import is_square
def A112886_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:all(d==1 or not is_square((d<<3)+1) for d in divisors(n,generator=True)), count(max(startvalue,1)))
A112886_list = list(islice(A112886_gen(),40)) # Chai Wah Wu, Jun 07 2025

Formula

a(n) = A132895(n)/2. - Ray Chandler, May 29 2008

Extensions

More terms from Farideh Firoozbakht, Jan 12 2006

Name edited (based on a suggestion from Michel Marcus) by Jon E. Schoenfield, Jul 02 2021