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

1,65

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 == 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)

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 *)

Prog

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

Crossrefs

Cf. A001221, A005091, A005094.

Cf. A079260, A027748, A004144, A009003.

Sequence in context: A320005 A325414 A216510 * A364127 A340999 A119395

Adjacent sequences: A005086 A005087 A005088 * A005090 A005091 A005092

Keyword

nonn

Author

N. J. A. Sloane

Status

approved