A005089 Number of distinct primes == 1 (mod 4) dividing n.

0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 2, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 2, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 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, A005091, A005094, A083025 (with multiplicity).

Cf. A079260, A027748, A004144, A009003.

Programs

Haskell

a005089 = sum . map a079260 . a027748_row
-- Reinhard Zumkeller, Jan 07 2013

Magma

[#[p:p in PrimeDivisors(n)|p mod 4 eq 1]: n in [1..100]]; // Marius A. Burtea, Jan 16 2020

Maple

A005089 := proc(n)
    local a,pe;
    a := 0 ;
    for pe in ifactors(n)[2] do
        if modp(op(1,pe),4) =1 then
            a := a+1 ;
        end if;
    end do:
    a ;
end  proc:
seq(A005089(n),n=1..100) ; # R. J. Mathar, Jul 22 2021

Mathematica

f[n_]:=Length@Select[If[n==1,{},FactorInteger[n]],Mod[#[[1]],4]==1&]; Table[f[n],{n,102}] (* Ray Chandler, Dec 18 2011 *)
a[n_] := DivisorSum[n, Boole[PrimeQ[#] && Mod[#, 4] == 1]&]; Array[a, 100] (* Jean-François Alcover, Dec 01 2015 *)

PARI

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

Formula

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

From Reinhard Zumkeller, Jan 07 2013: (Start)

a(n) = Sum_{k=1..A001221(n)} A079260(A027748(n,k)).

a(A004144(n)) = 0.

a(A009003(n)) > 0. (End)