A128905 - OEIS

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

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OFFSET

1,1

COMMENTS

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

LINKS

Table of n, a(n) for n=1..54.

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.

EXAMPLE

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.

MATHEMATICA

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

CROSSREFS

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

Sequence in context: A345124 A049390 A220040 * A276135 A350115 A211143

Adjacent sequences: A128902 A128903 A128904 * A128906 A128907 A128908

KEYWORD

nonn

AUTHOR

Zak Seidov, Apr 22 2007

STATUS

approved