A069903 Number of distinct prime factors of n-th triangular number.

0, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 3, 3, 2, 2, 2, 3, 4, 3, 2, 3, 3, 2, 2, 3, 3, 3, 3, 2, 3, 3, 3, 4, 3, 2, 3, 4, 3, 3, 3, 3, 4, 3, 2, 3, 3, 2, 3, 4, 3, 2, 3, 4, 4, 3, 2, 4, 4, 2, 3, 3, 3, 4, 3, 3, 4, 4, 3, 3, 3, 2, 3, 4, 4, 4, 3, 3, 3, 2, 2, 4, 5, 3, 3, 4, 3, 3, 4

Offset: 1

Author

Reinhard Zumkeller, Apr 10 2002

Keywords

• nonn
• easy

Examples

			A000217(11) = 11*(11+1)/2 = 66 = 2*3*11, therefore a(11) = 3.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000217, A001221, A059957, A069901, A069902, A069904, A077761.

Programs

Mathematica

PrimeNu[#]&/@Accumulate[Range[90]] (* Harvey P. Dale, Oct 06 2016 *)

PARI

a(n) = omega(n*(n+1)/2); \\ Michel Marcus, Feb 05 2021

PARI

a(n)=onega(n/gcd(n,2))+omega((n+1)/gcd(n+1)) \\ Charles R Greathouse IV, Sep 21 2024

Formula

a(n) = A001221(A000217(n)).

Sum_{k=1..n} a(k) = 2 * n * (log(log(n)) + B - 1/4) + O(n/log(n)), where B is Mertens's constant (A077761). - Amiram Eldar, Sep 21 2024