A318682 - OEIS

%I #28 Feb 16 2025 08:33:56

%S -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,

%T 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,

%U 15,14,15,16,17,16,15,14,15,16,15,14,15,14,15,16,17,18,17,16,17,18

%N 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.

%C 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.

%H Daniel Blaine McBride, <a href="/A318682/b318682.txt">Table of n, a(n) for n = 1..100000</a>

%H K. Alladi and P. Erdős, <a href="http://projecteuclid.org/euclid.pjm/1102811427">On an additive arithmetic function</a>, Pacific J. Math., Volume 71, Number 2 (1977), 275-294.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SumofPrimeFactors.html">Sum of Prime Factors</a>

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

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

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

%o (Python)

%o from sympy import factorint

%o def A318682(n):

%o ....a_n = 0

%o ....for i in range(1, n+1):

%o ........a_n += (-1)**(sum(p*e for p, e in factorint(i).items())+1)

%o ....return a_n

%o (PARI) sopfr(n) = my(f=factor(n)); sum(k=1, #f~, f[k, 1]*f[k, 2]);

%o a(n) = sum(k=1, n, (-1)^(sopfr(k)+1)); \\ _Michel Marcus_, Sep 09 2018

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

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

%K sign,changed

%O 1,2

%A _Daniel Blaine McBride_, Aug 30 2018