A209310 -id:A209310 - cp's OEIS frontend

A209309 Numbers whose sum of triangular divisors is also triangular and greater than 1.

6, 12, 18, 24, 48, 54, 96, 102, 110, 114, 138, 162, 174, 186, 192, 204, 220, 222, 228, 246, 258, 282, 315, 318, 348, 354, 364, 366, 372, 384, 402, 414, 426, 438, 440, 444, 456, 474, 486, 492, 498, 516, 522, 534, 550, 558, 564, 582, 606, 618, 636, 642, 654, 678

Offset: 1

Antonio Roldán, Jan 18 2013

nonn

			186 is a term because the sum of its triangular divisors, 1+3+6 = 10 is also triangular.

Antonio Roldán, Table of n, a(n) for n = 1..8055 (terms <= 10^5)

Cf. A000217, A035316, A185027, A209310.

triQ[n_] := n > 1 && IntegerQ[Sqrt[8*n+1]]; q[n_] := triQ[1 + DivisorSum[n, #&, triQ[#] &]]; Select[Range[700], q] (* Amiram Eldar, Aug 12 2023 *)

istriangular(n)=issquare(8*n+1)
{t=0; for(n=1, 10^5, k=sumdiv(n, d, istriangular(d)*d); if(istriangular(k)&&k>>1, t+=1; write("b209309.txt",t," ",n)))}

A209311 Numbers whose sum of triangular divisors is also a divisor and greater than 1.

Original entry on oeis.org

285, 1302, 1425, 1820, 2508, 3640, 3720, 4845, 4956, 5016, 5415, 7125, 7280, 9100, 9114, 9912, 11685, 12255, 12740, 14508, 15105, 16815, 17385, 18200, 19095, 19824, 20235, 20805, 22134, 22515, 23655, 23660, 24021, 24738, 25365, 25480, 27075, 27588, 27645

Offset: 1

Antonio Roldán, Jan 18 2013

nonn

			285 is in the sequence because its divisors being 1, 3, 5, 15, 19, 57, 95, 285, of which 1, 3 and 15 are triangular, these add up to 19.
1302 is in sequence because the sum of triangular divisors 21 + 6 + 3 + 1 = 31 is divisor of 1302.

Antonio Roldán, Table of n, a(n) for n = 1..13202 (terms < 10^7)

Cf. A185027, A035316, A209309, A209310.

TriangularQ[n_] := IntegerQ[Sqrt[1 + 8*n]]; fQ[n_] := Module[{tri = Total[Select[Divisors[n], TriangularQ]]}, tri > 1 && Mod[n, tri] == 0]; Select[Range[28000], fQ] (* T. D. Noe, Jan 23 2013 *)

istriangular(n)=issquare(8*n+1)
{t=0; for(n=1, 10^7, k=sumdiv(n, d, istriangular(d)*d); if(n/k==n\k&&k>>1, t+=1; write("b209311.txt",t," ",n)))}

cp's OEIS Frontend

A209309 Numbers whose sum of triangular divisors is also triangular and greater than 1.

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