A065339 Number of primes congruent to 3 modulo 4 dividing n (with multiplicity).

0, 0, 1, 0, 0, 1, 1, 0, 2, 0, 1, 1, 0, 1, 1, 0, 0, 2, 1, 0, 2, 1, 1, 1, 0, 0, 3, 1, 0, 1, 1, 0, 2, 0, 1, 2, 0, 1, 1, 0, 0, 2, 1, 1, 2, 1, 1, 1, 2, 0, 1, 0, 0, 3, 1, 1, 2, 0, 1, 1, 0, 1, 3, 0, 0, 2, 1, 0, 2, 1, 1, 2, 0, 0, 1, 1, 2, 1, 1, 0, 4, 0, 1, 2, 0, 1, 1, 1, 0, 2, 1, 1, 2, 1, 1, 1, 0, 2, 3, 0, 0, 1, 1, 0, 2

Offset: 1

Author

Reinhard Zumkeller, Oct 29 2001

Keywords

• nonn

Links

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

Crossrefs

Cf. A001222, A007814, A065338, A005091, A007949, A083025 (analogous for 4k+1 primes), A097706, A378194.

Cf. A020639, A027746, A028234, A032742, A067029, A260728, A260730.

Programs

Haskell

a065339 1 = 0
a065339 n = length [x | x <- a027746_row n, mod x 4 == 3]
-- Reinhard Zumkeller, Jan 10 2012

Maple

A065339 := proc(n)
    a := 0 ;
    for f in ifactors(n)[2] do
        if op(1,f) mod 4 = 3 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]==3&]; Table[f[n],{n,100}] (* Ray Chandler, Dec 18 2011 *)

PARI

A065339(n)=sum(i=1,#n=factor(n)~,if(n[1,i]%4==3,n[2,i]))  \\ M. F. Hasler, Apr 16 2012

Scheme

;; using memoization-macro definec
(definec (A065339 n) (cond ((< n 3) 0) ((even? n) (A065339 (/ n 2))) (else (+ (/ (- (modulo (A020639 n) 4) 1) 2) (A065339 (A032742 n))))))
;; Antti Karttunen, Aug 14 2015

Scheme

;; using memoization-macro definec
(definec (A065339 n) (cond ((< n 3) 0) ((even? n) (A065339 (/ n 2))) ((= 1 (modulo (A020639 n) 4)) (A065339 (A032742 n))) (else (+ (A067029 n) (A065339 (A028234 n))))))
;; Antti Karttunen, Aug 14 2015

Formula

a(n) = A001222(n) - A007814(n) - A083025(n).

(2^A007814(n)) * (3^a(n)) = A065338(n).

From Antti Karttunen, Aug 14 2015: (Start)

a(1) = a(2) = 0; thereafter, if n is even, a(n) = a(n/2), otherwise a(n) = ((A020639(n) mod 4)-1)/2 + a(n/A020639(n)). [Where A020639(n) gives the smallest prime factor of n.]

Other identities and observations. For all n >= 1:

a(n) = A007949(A065338(n)).

a(n) = A001222(A097706(n)).

a(n) >= A260728(n). [See A260730 for the positions of differences.] (End)

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