A205745 a(n) = card { d | d*p = n, d odd, p prime }

0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 0, 1, 1, 1, 0, 2, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 2, 1, 2, 0, 1, 1, 2, 0, 1, 1, 1, 0, 2, 1, 1, 0, 1, 1, 2, 0, 1, 1, 2, 0, 2, 1, 1, 0, 1, 1, 2, 0, 2, 1, 1, 0, 2, 1, 1, 0, 1, 1, 2, 0, 2, 1, 1, 0, 1, 1, 1, 0, 2, 1, 2

Offset

1,15

Comments

Equivalently, a(n) is the number of prime divisors p|n such that n/p is odd. - Gus Wiseman, Jun 06 2018

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Formula

O.g.f.: Sum_{p prime} x^p/(1 - x^(2p)). - Gus Wiseman, Jun 06 2018

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

Mathematica

a[n_] := Sum[ Boole[ OddQ[d] && PrimeQ[n/d] ], {d, Divisors[n]} ]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jun 27 2013 *)

Prog

(Sage)
def A205745(n):
    return sum((n//d) % 2 for d in divisors(n) if is_prime(d))
[A205745(n) for n in (1..105)]
(PARI) a(n)=if(n%2, omega(n), n%4/2) \\ Charles R Greathouse IV, Jan 30 2012
(Haskell)
a205745 n = sum $ map ((`mod` 2) . (n `div`))
   [p | p <- takeWhile (<= n) a000040_list, n `mod` p == 0]
-- Reinhard Zumkeller, Jan 31 2012

Crossrefs

Cf. A000005, A000607, A001221, A008683, A010051, A068050, A077761, A083399, A088705, A106404, A305614.

Sequence in context: A211159 A347440 A088434 * A333781 A243223 A034178

Adjacent sequences: A205742 A205743 A205744 * A205746 A205747 A205748

Keyword

nonn

Author

Peter Luschny, Jan 30 2012

Status

approved