A076578 -id:A076578 - cp's OEIS frontend

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

Jonathan Vos Post, Dec 14 2004

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

			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.

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.

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.

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

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

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

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

More terms from Ray Chandler, Dec 14 2004

A333771 Triangular numbers that are the product of four distinct primes.

Original entry on oeis.org

210, 1326, 1770, 1830, 2145, 2346, 2415, 2926, 3003, 3486, 4186, 4278, 5565, 6105, 6555, 6670, 7626, 8385, 8646, 9730, 11935, 12246, 13695, 16653, 17205, 17391, 17578, 18915, 22155, 22578, 24531, 25878, 26106, 27730, 27966, 28203, 30381, 32385, 33411, 35245

Offset: 1

Jon E. Schoenfield, Apr 04 2020

nonn

The maximum exponent for each prime in the factorization of each term is one. - Harvey P. Dale, Jul 21 2021

			The 20th triangular number, T(20) = 20*21/2 = 210 = 2 * 3 * 5 * 7, so 210 is a term.
T(1333) = 889111 = 23 * 29 * 31 * 43, so 889111 is a term.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000217 (triangular numbers), A068443 (triangular numbers that are the product of 2 distinct primes), A128896 (triangular numbers that are the product of 3 distinct primes).

Cf. A076578, A121479.

q:= n-> map(i-> i[2], ifactors(n)[2])=[1$4]:
select(q, [seq(n*(n+1)/2, n=0..300)])[];  # Alois P. Heinz, Apr 04 2020

Select[Accumulate[Range[300]],PrimeNu[#]==PrimeOmega[#]==4&] (* Harvey P. Dale, Jul 21 2021 *)

cp's OEIS Frontend

A101744 Triangular numbers which are 10-almost primes.

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