A079635 Sum of (2 - p mod 4) for all prime factors p of n (with repetition).

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

Offset: 1

Author

Reinhard Zumkeller, Jan 30 2003

Keywords

• sign

Comments

a(n) = {number of primes of the form 4k+1 dividing n} minus {number of primes of the form 4k+3 dividing n}, both counted with multiplicity. - Antti Karttunen, Feb 03 2016, after the formula.

Examples

			a(55) = a(5*11) = (2 - 5 mod 4)+(2 - 11 mod 4) = (2-1)+(2-3) = (1)+(-1) = 0.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A065339, A083025.

Cf. A072202 (indices of zeros), A268379 (of strictly positive terms), A268380 (of negative terms), A268381 (of nonnegative terms).

Cf. A027746, A267099.

Cf. A005094 (difference when counting only distinct primes).

Programs

Haskell

a079635 1 = 0
a079635 n = sum $ map ((2 - ) . (`mod` 4)) $ a027746_row n
-- Reinhard Zumkeller, Jan 10 2012

Maple

f:= proc(n) local t;
add(t[2]*(2-(t[1] mod 4)), t=ifactors(n)[2])
end proc:
map(f, [$1..100]); # Robert Israel, Feb 05 2016

Mathematica

f[n_]:=Plus@@((2-Mod[#[[1]],4])*#[[2]]&/@If[n==1,{},FactorInteger[n]]); Table[f[n],{n,100}] (* Ray Chandler, Dec 20 2011 *)

Scheme

(define (A079635 n) (- (A083025 n) (A065339 n))) ;; Antti Karttunen, Feb 03 2016

Formula

a(n) = A083025(n) - A065339(n).

Other identities. For all n >= 1:

a(A267099(n)) = -a(n). - Antti Karttunen, Feb 03 2016

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

Extensions

Edited by Ray Chandler, Dec 20 2011