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

Author

N. J. A. Sloane

Keywords

• nonn

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.

Crossrefs

Cf. A001221, A005089, A005094.

Cf. A079261, A027748, A072437, A187811.

Programs

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

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 *)

PARI

for(n=1,100,print1(sumdiv(n,d,isprime(d)*if((d-3)%4,0,1)),","))

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

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