A083025 Number of primes congruent to 1 modulo 4 dividing n (with multiplicity).

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

Offset: 1

Author

Reinhard Zumkeller, Oct 29 2001

Keywords

• nonn
• easy
• nice

References

• David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989, p. 61.

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000

Crossrefs

First differs from A046080 at n=65.

Cf. A001222, A007814, A027746, A065339 (== 3 (mod 4)), A378879 (=2,3 (mod 4)), A005089 (without multiplicity).

Programs

Haskell

a083025 1 = 0
a083025 n = length [x | x <- a027746_row n, mod x 4 == 1]
-- Reinhard Zumkeller, Jan 10 2012

Maple

A083025 := proc(n)
    a := 0 ;
    for f in ifactors(n)[2] do
        if op(1,f) mod 4 = 1 then
            a := a+op(2,f) ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Dec 16 2011

Mathematica

f[n_]:=Plus@@Last/@Select[If[n==1,{},FactorInteger[n]],Mod[#[[1]],4]==1&]; Table[f[n],{n,100}] (* Ray Chandler, Dec 18 2011 *)

PARI

A083025(n)=sum(i=1,#n=factor(n)~,if(n[1,i]%4==1,n[2,i]))  \\ M. F. Hasler, Apr 16 2012

Formula

a(n) = A001222(n) - A007814(n) - A065339(n).

Totally additive with a(2) = 0, a(p) = 1 if p == 1 (mod 4), and a(p) = 0 if p == 3 (mod 4). - Amiram Eldar, Jun 17 2024