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A128896
Triangular numbers that are products of three distinct primes.
8
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
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
KEYWORD
nonn
AUTHOR
Zak Seidov, Apr 20 2007
EXTENSIONS
Name clarified by Tanya Khovanova, Sep 06 2022
STATUS
approved