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%I #13 Sep 06 2022 14:43:16
%S 66,78,105,190,231,406,435,465,561,595,741,861,903,946,1378,1653,2211,
%T 2278,2485,3081,3655,3741,4371,4465,5151,5253,5995,6441,7021,7503,
%U 8515,8911,9453,9591,10011,10153,10585,11026,12561,13366,14878,15051,15753
%N Triangular numbers that are products of three distinct primes.
%H Harvey P. Dale, <a href="/A128896/b128896.txt">Table of n, a(n) for n = 1..1000</a>
%F 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.
%F Equals A000217 INTERSECT A007304 and A075875 INTERSECT A121478. - _R. J. Mathar_, Apr 22 2007
%e 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.
%e 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.
%t Select[Table[n(n+1)/2,{n,1,210}],Transpose[FactorInteger[ # ]][[2]]=={1,1,1}&]
%t Select[Accumulate[Range[200]],PrimeNu[#]==PrimeOmega[#]==3&] (* _Harvey P. Dale_, Apr 23 2017 *)
%Y Cf. A000217, A068443, A069903, A076551, A127637.
%K nonn
%O 1,1
%A _Zak Seidov_, Apr 20 2007
%E Name clarified by _Tanya Khovanova_, Sep 06 2022
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