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A163301 a(n) = Sum_{x=n-th even nonprime..n-th odd nonprime} -x*(-1)^x.

%I A163301 #18 May 30 2025 15:49:41
%S A163301 1,3,5,7,8,8,10,10,11,13,14,14,15,15,17,17,18,20,20,21,22,22,23,23,23,
%T A163301 24,26,28,29,29,29,29,29,29,30,31,31,33,33,33,33,35,35,36,36,37,38,38,
%U A163301 39,39,41,41,41,41,43,45,45,45,45,45,46,46,46,46,46,47,49,50,50,52,52
%N A163301 a(n) = Sum_{x=n-th even nonprime..n-th odd nonprime} -x*(-1)^x.
%C A163301 Here n-th even nonprime = A163300(n), n-th odd nonprime = A014076(n) and A163300 U A014076 = A141468.
%C A163301 Seems to be essentially the same as A008508. - _R. J. Mathar_, May 30 2025
%F A163301 a(n) = Sum_{x=A163300(n)..A014076(n)}-x*(-1)^x.
%F A163301 a(n) = A001057( A014076(n)) - A001057(A163300(n)-1). - _R. J. Mathar_, May 21 2010
%e A163301 a(1) = -0*(-1)^0 - 1*(-1)^1 = 0 + 1 = 1;
%e A163301 a(2) = -4*(-1)^4 - 5*(-1)^5 - 6*(-1)6 - 7*(-1)^7 - 8*(-1)^8 - 9*(-1)^9 = -4 + 5 - 6 + 7 - 8 + 9 = 3.
%p A163301 A163300 := proc(n) if n <= 2 then op(n,[0,4]) ; else for a from procname(n-1)+2 by 2 do if not isprime(a) then return a; end if; end do; end if; end proc:
%p A163301 A014076 := proc(n) if n = 1 then 1; else for a from procname(n-1)+2 by 2 do if not isprime(a) then return a ; end if; end do: end if; end proc:
%p A163301 A001057 := proc(n) (1-(-1)^n*(2*n+1))/4; end proc:
%p A163301 A163301 := proc(n) A001057( A014076(n)) - A001057(A163300(n)-1) ; end proc: seq(A163301(n),n=1..120) ; # _R. J. Mathar_, May 21 2010
%Y A163301 Cf. A014076, A141468, A163300.
%K A163301 nonn
%O A163301 1,2
%A A163301 _Juri-Stepan Gerasimov_, Jul 26 2009
%E A163301 Corrected from a(39) onwards by _R. J. Mathar_, May 21 2010