cp's OEIS Frontend

A325978 a(n) = (1/2)*(A325314(n) + A325814(n)).

%I A325978 #21 Feb 22 2024 02:17:24
%S A325978 1,2,3,1,5,6,7,-1,3,10,11,0,13,14,15,-5,17,0,19,2,21,22,23,-12,10,26,
%T A325978 3,4,29,30,31,-13,33,34,35,-24,37,38,39,-14,41,42,43,8,9,46,47,-36,21,
%U A325978 5,51,10,53,-18,55,-16,57,58,59,-12,61,62,15,-29,65,66,67,14,69,70,71,-72,73,74,15,16,77,78,79,-46,3,82,83,-12,85
%N A325978 a(n) = (1/2)*(A325314(n) + A325814(n)).
%C A325978 Question: Are a(12) = 0 and a(18) = 0 the only zeros in this sequence?
%H A325978 Antti Karttunen, <a href="/A325978/b325978.txt">Table of n, a(n) for n = 1..20000</a>
%F A325978 a(n) = (1/2)*(A325314(n) + A325814(n)).
%F A325978 a(n) = n - A325974(n).
%F A325978 a(n) = A033879(n) + A325977(n).
%F A325978 Sum_{k=1..n} a(k) ~ c * n^2, where c = 3/4 - zeta(2)*(1/2 - 1/(4*zeta(3))) = 0.2696411609... . - _Amiram Eldar_, Feb 22 2024
%t A325978 Table[(1/2) If[n == 1, 2, 2 n - DivisorSigma[1, n] + Times @@ (1 + FactorInteger[n][[;; , 1]]) - DivisorSum[n, # &, ! CoprimeQ[#, n/#] &]], {n, 85}] (* _Michael De Vlieger_, Jun 06 2019 *)
%o A325978 (PARI)
%o A325978 A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
%o A325978 A325314(n) = (n - A162296(n));
%o A325978 A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
%o A325978 A048146(n) = (sigma(n)-A034448(n));
%o A325978 A325814(n) = (n-A048146(n));
%o A325978 A325978(n) = ((A325314(n)+A325814(n))/2);
%Y A325978 Cf. A000203, A033879, A034448, A048146, A162296, A325314, A325814, A325973, A325974, A325975, A325977, A325979, A325981.
%Y A325978 Cf. A002117, A013661.
%K A325978 sign
%O A325978 1,2
%A A325978 _Antti Karttunen_, Jun 02 2019