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 A101744 Triangular numbers which are 10-almost primes. 2

%I

%S 32640,73920,130816,165600,204480,265356,294528,401856,592416,839160,

%T 947376,990528,1279200,1445850,1492128,1606528,1842240,1844160,

%U 2031120,2049300,2821500,2956096,3571128,3963520,4148640,4250070,4335040

%N Triangular numbers which are 10-almost primes.

%C A101745 contains the indices of this sequence, i.e. T(n) for what values of n are these 10-almost primes. Mathematica code and extension by _Ray Chandler_.

%D Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 59, 1987.

%D Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 33-38, 1996.

%D Dudeney, H. E. Amusements in Mathematics. New York: Dover, pp. 67 and 167, 1970.

%H Charles R Greathouse IV, <a href="/A101744/b101744.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TriangularNumber.html">Triangular Number</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AlmostPrime.html">Almost Prime</a>.

%F a(n) is in the intersection of {A000217} and {A046314}. Integers of the form k*(k+1)/2 which have exactly 10 prime factors.

%e a(1) = 32640 because that is the smallest triangular number which is also a 10-almost prime; specifically T(255) = 255*(255+1)/2 = 32640 = 2^7 * 3 * 5 * 17.

%t BigOmega[n_Integer]:=Plus@@Last[Transpose[FactorInteger[n]]]; Select[Table[n*(n+1)/2, {n, 2, 5000}], BigOmega[ # ]?10&]

%o (PARI) list(lim)=my(v=List(),cur,last=3,n=256,t); while((t=n*(n-1)/2) <= lim, cur=bigomega(n); if(cur+old==11, listput(v,t)); old=cur; n++); Vec(v) \\ _Charles R Greathouse IV_, Feb 05 2017

%Y Cf. A000217, A046314, A068443, A075875, A076578-A076583, A075088, A101745.

%K easy,nonn

%O 1,1

%A _Jonathan Vos Post_, Dec 14 2004

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