A318682 a(n) is the number of odd values minus the number of even values of the integer log of all positive integers up to and including n.

-1, -2, -1, -2, -1, 0, 1, 0, -1, 0, 1, 2, 3, 4, 3, 2, 3, 2, 3, 4, 3, 4, 5, 6, 5, 6, 7, 8, 9, 8, 9, 8, 7, 8, 7, 6, 7, 8, 7, 8, 9, 8, 9, 10, 11, 12, 13, 14, 13, 12, 11, 12, 13, 14, 13, 14, 13, 14, 15, 14, 15, 16, 17, 16, 15, 14, 15, 16, 15, 14, 15, 14, 15, 16, 17, 18, 17, 16, 17, 18

Offset: 1

Author

Daniel Blaine McBride, Aug 30 2018

Keywords

• sign

Comments

a(n) = Sum_{k=1..n} (-1)^(sopfr(k)+1), with sopfr(n) the sum of the prime factors of n with repetition, also known as the integer log of n.

Examples

			a(4) = -1 - 1 + 1 - 1 = -2, since sopfr(1) = 0, sopfr(2) = 2, sopfr(3) = 3, and sopfr(4) = 4.
		

Links

• Daniel Blaine McBride, Table of n, a(n) for n = 1..100000
• K. Alladi and P. Erdős, On an additive arithmetic function, Pacific J. Math., Volume 71, Number 2 (1977), 275-294.
• Eric Weisstein's World of Mathematics, Sum of Prime Factors

Crossrefs

Cf. A001414 (sum of prime divisors of n with repetition, sopfr(n)).

Cf. A036349 (numbers such that sopfr(n) is even).

Programs

Mathematica

Nest[Append[#, #[[-1]] + (-1)^(1 + Total@ Flatten[ConstantArray[#1, #2] & @@@ FactorInteger[Length@ # + 1] ])] &, {-1}, 79] (* Michael De Vlieger, Sep 10 2018 *)

PARI

sopfr(n) = my(f=factor(n)); sum(k=1, #f~, f[k, 1]*f[k, 2]);
a(n) = sum(k=1, n, (-1)^(sopfr(k)+1)); \\ Michel Marcus, Sep 09 2018

Python

from sympy import factorint
def A318682(n):
    a_n = 0
    for i in range(1, n+1):
        a_n += (-1)**(sum(p*e for p, e in factorint(i).items())+1)
    return a_n

Formula

a(n) = a(n-1) + (-1)^(sopfr(n)+1) with a(1) = (-1)^(sopfr(1)+1) = -1.