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

1,3

Links

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

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

Example

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

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A272231 A209253 A165113 * A331003 A086007 A108582

Adjacent sequences: A069900 A069901 A069902 * A069904 A069905 A069906

Keyword

nonn,easy

Author

Reinhard Zumkeller, Apr 10 2002

Status

approved