A101745 Indices of triangular numbers which are 10-almost primes. Indices of A101744.
255, 384, 511, 575, 639, 728, 767, 896, 1088, 1295, 1376, 1407, 1599, 1700, 1727, 1792, 1919, 1920, 2015, 2024, 2375, 2431, 2672, 2815, 2880, 2915, 2944, 2975, 3104, 3159, 3199, 3327, 3375, 3392, 3456, 3583, 3744, 3999, 4031, 4032, 4160, 4223, 4256
Offset: 1
Examples
a(1) = 255 because that is the smallest index of a triangular number which is also a 10-almost prime; specifically T(255) = 255*(255+1)/2 = 32640 = 2^7 * 3 * 5 * 17.
Links
- Muniru A Asiru, Table of n, a(n) for n = 1..10000
Programs
-
GAP
F:=List([1..4300],n->Length(Factors(n*(n+1)/2)));; a:=Filtered([1..Length(F)],i->F[i]=10); # Muniru A Asiru, Dec 22 2018
-
Magma
[n: n in [2..4500] | &+[d[2]: d in Factorization((n*(n+1)))] eq 11]; // Vincenzo Librandi, Dec 22 2018
-
Mathematica
BigOmega[n_Integer]:=Plus@@Last[Transpose[FactorInteger[n]]]; Do[ t=n*(n+1)/2; If[BigOmega[t]==10, Print[n, " ", t];];, {n, 2, 5000}]; (* Ray Chandler, Dec 14 2004 *) Flatten[Position[Accumulate[Range[5000]],?(PrimeOmega[#]==10&)]] (* _Harvey P. Dale, May 12 2011 *)
Formula
a(n)*(a(n)+1)/2 has exactly 10 prime factors.
{ m : A069904(m) = 10 }. - Alois P. Heinz, Aug 05 2019
Extensions
More terms from Ray Chandler, Dec 14 2004