cp's OEIS Frontend

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.

%I A318682 #30 May 22 2025 10:21:48
%S A318682 -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 A318682 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 A318682 15,14,15,16,17,16,15,14,15,16,15,14,15,14,15,16,17,18,17,16,17,18
%N 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.
%C A318682 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 A318682 Daniel Blaine McBride, <a href="/A318682/b318682.txt">Table of n, a(n) for n = 1..100000</a>
%H A318682 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 A318682 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SumofPrimeFactors.html">Sum of Prime Factors</a>
%F A318682 a(n) = a(n-1) + (-1)^(sopfr(n)+1) with a(1) = (-1)^(sopfr(1)+1) = -1.
%e A318682 a(4) = -1 - 1 + 1 - 1 = -2, since sopfr(1) = 0, sopfr(2) = 2, sopfr(3) = 3, and sopfr(4) = 4.
%t A318682 Nest[Append[#, #[[-1]] + (-1)^(1 + Total@ Flatten[ConstantArray[#1, #2] & @@@ FactorInteger[Length@ # + 1] ])] &, {-1}, 79] (* _Michael De Vlieger_, Sep 10 2018 *)
%o A318682 (Python)
%o A318682 from sympy import factorint
%o A318682 def A318682(n):
%o A318682     a_n = 0
%o A318682     for i in range(1, n+1):
%o A318682         a_n += (-1)**(sum(p*e for p, e in factorint(i).items())+1)
%o A318682     return a_n
%o A318682 (PARI) sopfr(n) = my(f=factor(n)); sum(k=1, #f~, f[k, 1]*f[k, 2]);
%o A318682 a(n) = sum(k=1, n, (-1)^(sopfr(k)+1)); \\ _Michel Marcus_, Sep 09 2018
%Y A318682 Cf. A001414 (sum of prime divisors of n with repetition, sopfr(n)).
%Y A318682 Cf. A036349 (numbers such that sopfr(n) is even).
%K A318682 sign
%O A318682 1,2
%A A318682 _Daniel Blaine McBride_, Aug 30 2018