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A383641 a(n) is the difference between the sum of even composites and the sum of the odd composites in the first n positive integers.

%I A383641 #18 May 14 2025 21:44:46
%S A383641 0,0,0,4,4,10,10,18,9,19,19,31,31,45,30,46,46,64,64,84,63,85,85,109,
%T A383641 84,110,83,111,111,141,141,173,140,174,139,175,175,213,174,214,214,
%U A383641 256,256,300,255,301,301,349,300,350,299,351,351,405,350,406,349,407,407
%N A383641 a(n) is the difference between the sum of even composites and the sum of the odd composites in the first n positive integers.
%H A383641 Felix Huber, <a href="/A383641/b383641.txt">Table of n, a(n) for n = 1..10000</a>
%H A383641 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompositeNumber.html">Composite Number</a>
%F A383641 a(n) = floor((n-2)/2) - n*(n mod 2) + Sum_{i=2..pi(n)} prime(i) for n > 1.
%F A383641 a(n) = A004526(n) - A193356(n) - A010701(n) + A034387(A000720(n)) for n > 1.
%F A383641 a(n) = Sum_{i=1..n} ((-1)^i*i*A066247(i)).
%e A383641 Of the first 9 positive integers, 4, 6, and 8 are even composites and 9 is an odd composite, so a(9) = 4 + 6 + 8 - 9 = 9.
%p A383641 A383641:=n->`if`(n=1,0,floor((n-2)/2)-n*(n mod 2)+add(ithprime(i),i=2..NumberTheory:-pi(n)));seq(A383641(n),n=1..59);
%t A383641 lim=59;cn=Select[Range[lim],CompositeQ];a[n_]:=Total[Select[cn,EvenQ[#]&&#<=n&]]-Total[Select[cn,OddQ[#]&&#<=n&]];Array[a,lim] (* _James C. McMahon_, May 14 2025 *)
%Y A383641 Cf. A000720, A002808, A004526, A010701, A028552, A034387, A066247, A071904, A101256, A193356, A262044, A383259.
%K A383641 nonn
%O A383641 1,4
%A A383641 _Felix Huber_, May 08 2025