A128905 Numbers k such that the k-th triangular number has exactly four distinct prime factors.

20, 51, 59, 60, 65, 68, 69, 76, 77, 83, 91, 92, 105, 110, 114, 115, 123, 129, 131, 139, 154, 156, 165, 182, 185, 186, 187, 194, 210, 212, 221, 227, 228, 235, 236, 237, 246, 254, 258, 265, 266, 267, 273, 276, 286, 290, 291, 307, 309, 318, 321, 322, 330, 345

Offset: 1

Author

Zak Seidov, Apr 22 2007

Keywords

• nonn

Comments

Or, indices of triangular numbers with exactly four distinct prime factors.

Examples

			In order of increasing p (the least prime factor of T(k)):
  a(1)  =  20 because T(20)  =    210 =  2* 3* 5* 7,
  a(5)  =  65 because T(65)  =   2145 =  3* 5*11*13,
  a(21) = 154 because T(154) =  11935 =  5* 7*11*31,
  a(45) = 286 because T(286) =  41041 =  7*11*13*41,
  a(143)= 781 because T(781) = 305371 = 11*17*23*71,
  a(91) = 493 because T(493) = 121771 = 13*17*19*29, etc.
		

Crossrefs

Cf. A000217, A068443, A069903, A076551, A127637, A128896.

Programs

Mathematica

lim=346;tn=Rest[Array[ #*(# - 1)/2 &,lim]];Select[Range[lim-1],PrimeNu[tn[[#]]]==PrimeOmega[tn[[#]]]==4&] (* James C. McMahon, Jan 12 2025 *)

Formula

a(n)=k and T(k)=k*(k+1)/2=p*q*r*s for some k, p, q, r, s where T(k) is a triangular number and p, q, r, s are distinct primes.