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A079635
Sum of (2 - p mod 4) for all prime factors p of n (with repetition).
8
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
1,9
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.
LINKS
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
EXAMPLE
a(55) = a(5*11) = (2 - 5 mod 4)+(2 - 11 mod 4) = (2-1)+(2-3) = (1)+(-1) = 0.
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 *)
PROG
(Haskell)
a079635 1 = 0
a079635 n = sum $ map ((2 - ) . (`mod` 4)) $ a027746_row n
-- Reinhard Zumkeller, Jan 10 2012
(Scheme) (define (A079635 n) (- (A083025 n) (A065339 n))) ;; Antti Karttunen, Feb 03 2016
CROSSREFS
Cf. A072202 (indices of zeros), A268379 (of strictly positive terms), A268380 (of negative terms), A268381 (of nonnegative terms).
Cf. A005094 (difference when counting only distinct primes).
Sequence in context: A365126 A168509 A347822 * A037909 A181506 A319797
KEYWORD
sign
AUTHOR
Reinhard Zumkeller, Jan 30 2003
EXTENSIONS
Edited by Ray Chandler, Dec 20 2011
STATUS
approved