A101744 Triangular numbers which are 10-almost primes.

32640, 73920, 130816, 165600, 204480, 265356, 294528, 401856, 592416, 839160, 947376, 990528, 1279200, 1445850, 1492128, 1606528, 1842240, 1844160, 2031120, 2049300, 2821500, 2956096, 3571128, 3963520, 4148640, 4250070, 4335040

Offset: 1

Author

Jonathan Vos Post, Dec 14 2004

Keywords

• easy
• nonn

Comments

A101745 contains the indices of this sequence, i.e. T(n) for what values of n are these 10-almost primes.

Examples

			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.
		

References

• Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 59, 1987.
• Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 33-38, 1996.
• Dudeney, H. E. Amusements in Mathematics. New York: Dover, pp. 67 and 167, 1970.

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Triangular Number.
• Eric Weisstein's World of Mathematics, Almost Prime.

Crossrefs

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

Programs

Mathematica

BigOmega[n_Integer]:=Plus@@Last[Transpose[FactorInteger[n]]]; Select[Table[n*(n+1)/2, {n, 2, 5000}], BigOmega[ # ]==10&] (* Ray Chandler, Dec 14 2004 *)

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

Formula

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

Extensions

More terms from Ray Chandler, Dec 14 2004