A128896 Triangular numbers that are products of three distinct primes.

66, 78, 105, 190, 231, 406, 435, 465, 561, 595, 741, 861, 903, 946, 1378, 1653, 2211, 2278, 2485, 3081, 3655, 3741, 4371, 4465, 5151, 5253, 5995, 6441, 7021, 7503, 8515, 8911, 9453, 9591, 10011, 10153, 10585, 11026, 12561, 13366, 14878, 15051, 15753

Offset

1,1

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Formula

a(n) = T(k) = k*(k+1)/2 = p*q*r for some k,p,q,r, where T(k) is triangular number and p, q, r are distinct primes.

Equals A000217 INTERSECT A007304 and A075875 INTERSECT A121478. - R. J. Mathar, Apr 22 2007

Example

a(1)=T(11)=66=2*3*11, a(2)=T(12)=78=2*3*13, a(3)=T(14)=105=3*5*7, a(4)=T(19)=190=2*5*19, a(5)=T(21)=231=3*7*11, a(6)=T(28)=406=2*7*29.
T(15) = 120 = 2^3*3*5. The triangular 120 has three prime factors but is not a product of these factors. Thus, 120 is not in this sequence.

Mathematica

Select[Table[n(n+1)/2, {n, 1, 210}], Transpose[FactorInteger[ # ]][[2]]=={1, 1, 1}&]
Select[Accumulate[Range[200]], PrimeNu[#]==PrimeOmega[#]==3&] (* Harvey P. Dale, Apr 23 2017 *)

Crossrefs

Cf. A000217, A068443, A069903, A076551, A127637.

Sequence in context: A095751 A121478 A330809 * A109750 A127654 A293175

Adjacent sequences: A128893 A128894 A128895 * A128897 A128898 A128899

Keyword

nonn

Author

Zak Seidov, Apr 20 2007

Extensions

Name clarified by Tanya Khovanova, Sep 06 2022

Status

approved