A332423 If n = Product (p_j^k_j) then a(n) = Sum ((-1)^(k_j + 1) * k_j).

0, 1, 1, -2, 1, 2, 1, 3, -2, 2, 1, -1, 1, 2, 2, -4, 1, -1, 1, -1, 2, 2, 1, 4, -2, 2, 3, -1, 1, 3, 1, 5, 2, 2, 2, -4, 1, 2, 2, 4, 1, 3, 1, -1, -1, 2, 1, -3, -2, -1, 2, -1, 1, 4, 2, 4, 2, 2, 1, 0, 1, 2, -1, -6, 2, 3, 1, -1, 2, 3, 1, 1, 1, 2, -1, -1, 2, 3, 1, -3

Offset: 1

Author

Ilya Gutkovskiy, Feb 12 2020

Keywords

• sign
• easy

Comments

Sum of odd exponents in prime factorization of n minus the sum of even exponents in prime factorization of n.

Examples

			a(2700) = a(2^2 * 3^3 * 5^2) = -2 + 3 - 2 = -1.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Index entries for sequences computed from exponents in factorization of n.

Crossrefs

Cf. A001222, A037916, A067255, A071321, A077761, A195017, A316523, A319273, A332422, A350386, A350387.

Programs

Mathematica

a[n_] := Plus @@ ((-1)^(#[[2]] + 1) #[[2]] & /@ FactorInteger[n]); a[1] = 0; Table[a[n], {n, 1, 80}]

PARI

a(n) = vecsum(apply(x -> (-1)^(x+1) * x, factor(n)[, 2])); \\ Amiram Eldar, Oct 09 2023

Formula

From Amiram Eldar, Oct 09 2023: (Start)

Additive with a(p^e) = (-1)^(e+1) * e.

a(n) = A350387(n) - A350386(n).

Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B - C), where B is Mertens's constant (A077761) and C = Sum_{p prime} (3*p+1)/(p*(p+1)^2) = 0.81918453457738985491 ... . (End)