A069904 Number of prime factors of n-th triangular number (with multiplicity).

0, 1, 2, 2, 2, 2, 3, 4, 3, 2, 3, 3, 2, 3, 5, 4, 3, 3, 3, 4, 3, 2, 4, 5, 3, 4, 5, 3, 3, 3, 5, 6, 3, 3, 5, 4, 2, 3, 5, 4, 3, 3, 3, 5, 4, 2, 5, 6, 4, 4, 4, 3, 4, 5, 5, 5, 3, 2, 4, 4, 2, 4, 8, 7, 4, 3, 3, 4, 4, 3, 5, 5, 2, 4, 5, 4, 4, 3, 5, 8, 5, 2, 4, 5, 3, 3, 5, 4, 4, 5

Offset: 1

Author

Reinhard Zumkeller, Apr 10 2002

Keywords

• nonn

Examples

			A000217(8) = 8*(8+1)/2 = 36 = 2*2*3*3, therefore a(8) = 4.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A069901, A069902, A069903, A092405, A101745, A108815, A114435, A114436, A114437, A164977, A240527, A240528, A240529, A278253.

Programs

Mathematica

Array[Plus@@Last/@FactorInteger[ #*(#+1)/2]&,33] (* Vladimir Joseph Stephan Orlovsky, Feb 28 2010 *)

PARI

A069904(n) = bigomega((n*(n+1))/2); \\ Antti Karttunen, Oct 07 2017

Formula

a(n) = A001222(A000217(n)).

From Antti Karttunen, Oct 07 2017: (Start)

a(n) = (A001222(n)+A001222(n+1))-1.

a(n) = A001222(A278253(n)). (End)

From Alois P. Heinz, Aug 05 2019: (Start)

a(n) = 2 <=> n in { A164977 }.

a(n) = 3 <=> n in { A108815 }.

a(n) = 4 <=> n in { A114435 }.

a(n) = 5 <=> n in { A114436 }.

a(n) = 6 <=> n in { A114437 }.

a(n) = 7 <=> n in { A240527 }.

a(n) = 8 <=> n in { A240528 }.

a(n) = 9 <=> n in { A240529 }.

a(n) = 10 <=> n im { A101745 }. (End)