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

%I A079635 #30 Jun 17 2024 07:15:30
%S A079635 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,
%T A079635 -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,
%U A079635 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
%N A079635 Sum of (2 - p mod 4) for all prime factors p of n (with repetition).
%C A079635 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.
%H A079635 Reinhard Zumkeller, <a href="/A079635/b079635.txt">Table of n, a(n) for n = 1..10000</a>
%F A079635 a(n) = A083025(n) - A065339(n).
%F A079635 Other identities. For all n >= 1:
%F A079635 a(A267099(n)) = -a(n). - _Antti Karttunen_, Feb 03 2016
%F A079635 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
%e A079635 a(55) = a(5*11) = (2 - 5 mod 4)+(2 - 11 mod 4) = (2-1)+(2-3) = (1)+(-1) = 0.
%p A079635 f:= proc(n) local t;
%p A079635 add(t[2]*(2-(t[1] mod 4)), t=ifactors(n)[2])
%p A079635 end proc:
%p A079635 map(f, [$1..100]); # _Robert Israel_, Feb 05 2016
%t A079635 f[n_]:=Plus@@((2-Mod[#[[1]],4])*#[[2]]&/@If[n==1,{},FactorInteger[n]]); Table[f[n],{n,100}] (* _Ray Chandler_, Dec 20 2011 *)
%o A079635 (Haskell)
%o A079635 a079635 1 = 0
%o A079635 a079635 n = sum $ map ((2 - ) . (`mod` 4)) $ a027746_row n
%o A079635 -- _Reinhard Zumkeller_, Jan 10 2012
%o A079635 (Scheme) (define (A079635 n) (- (A083025 n) (A065339 n))) ;; _Antti Karttunen_, Feb 03 2016
%Y A079635 Cf. A065339, A083025.
%Y A079635 Cf. A072202 (indices of zeros), A268379 (of strictly positive terms), A268380 (of negative terms), A268381 (of nonnegative terms).
%Y A079635 Cf. A027746, A267099.
%Y A079635 Cf. A005094 (difference when counting only distinct primes).
%K A079635 sign
%O A079635 1,9
%A A079635 _Reinhard Zumkeller_, Jan 30 2003
%E A079635 Edited by _Ray Chandler_, Dec 20 2011