A059957 -id:A059957 - cp's OEIS frontend

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

Reinhard Zumkeller, Apr 10 2002

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

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

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

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

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

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

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

A192688 Sum of omega-values for two consecutive indices where they equal the omega-value at the sum of the two indices.

Original entry on oeis.org

1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 4, 3, 3

Offset: 1

Juri-Stepan Gerasimov, Jul 07 2011

nonn

The sequence contains sums A001221(n) + A001221(n+1) whenever they are equal to A001221(2n+1).

Generated by indices n = 1, 7, 16, 31, 52, 82, 97, 136, 157, 172, 178, 192, 199, 232, 241, 256, 262, 277, 292, ...

Cf. A001221, A059957, A092523.

for n from 1 to 2000 do
        if A001221(n)+A001221(n+1) = A001221(2*n+1) then
                printf("%d,",A001221(2*n+1)) ;
        end if;
end do: # R. J. Mathar, Oct 12 2011

f[n_] := Block[{a = PrimeNu[n] + PrimeNu[n + 1]}, If[a == PrimeNu[2n + 1], Return@ a]]; k = 1; lst = {}; While[k < 2600, If[f@k > 0, AppendTo[lst, f@ k]]; k++]; lst (* Robert G. Wilson v, Aug 29 2011 *)

cp's OEIS Frontend

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

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