A005091 Number of distinct primes = 3 mod 4 dividing n.

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

Offset

1,21

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Étienne Fouvry and Peter Koymans, On Dirichlet biquadratic fields, arXiv:2001.05350 [math.NT], 2020.

Formula

Additive with a(p^e) = 1 if p = 3 (mod 4), 0 otherwise.

From Reinhard Zumkeller, Jan 07 2013: (Start)

a(n) = Sum_{k=1..A001221(n)} A079261(A027748(n,k)).

a(A072437(n)) = 0.

a(A187811(n)) > 0. (End)

a(n) = Sum_{p|n} (binomial(p,3) mod 2), where p is a prime. - Ridouane Oudra, Nov 19 2019

Maple

with(numtheory): seq(add(binomial(p, 3) mod 2, p in factorset(n)), n=1..100); # Ridouane Oudra, Nov 19 2019

Mathematica

f[n_]:=Length@Select[If[n==1, {}, FactorInteger[n]], Mod[#[[1]], 4]==3&]; Table[f[n], {n, 102}] (* Ray Chandler, Dec 18 2011 *)

Prog

(PARI) for(n=1, 100, print1(sumdiv(n, d, isprime(d)*if((d-3)%4, 0, 1)), ", "))
(Haskell)
a005091 = sum . map a079261 . a027748_row
-- Reinhard Zumkeller, Jan 07 2013
(Magma) [0] cat [#[p:p in PrimeDivisors(n)| p mod 4 eq 3]: n in [2..100]]; // Marius A. Burtea, Nov 19 2019
(Magma) [0] cat [&+[Binomial(p, 3) mod 2:p in PrimeDivisors(n)]:n in [2..100]]; // Marius A. Burtea, Nov 19 2019
(Python)
from sympy import primefactors
def A005091(n): return sum(1 for p in primefactors(n) if p&3==3) # Chai Wah Wu, Jul 07 2024

Crossrefs

Cf. A001221, A005089, A005094.

Cf. A079261, A027748, A072437, A187811.

Sequence in context: A277778 A255319 A214088 * A353455 A276516 A346100

Adjacent sequences: A005088 A005089 A005090 * A005092 A005093 A005094

Keyword

nonn

Author

N. J. A. Sloane

Status

approved