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A325974 Arithmetic mean of {sum of non-unitary divisors} and {sum of nonsquarefree divisors}: a(n) = (1/2)*(A048146(n) + A162296(n)).

%I A325974 #18 Feb 22 2024 02:17:59
%S A325974 0,0,0,3,0,0,0,9,6,0,0,12,0,0,0,21,0,18,0,18,0,0,0,36,15,0,24,24,0,0,
%T A325974 0,45,0,0,0,60,0,0,0,54,0,0,0,36,36,0,0,84,28,45,0,42,0,72,0,72,0,0,0,
%U A325974 72,0,0,48,93,0,0,0,54,0,0,0,144,0,0,60,60,0,0,0,126,78,0,0,96,0,0,0,108,0,108,0,72,0,0,0,180,0,84,72
%N A325974 Arithmetic mean of {sum of non-unitary divisors} and {sum of nonsquarefree divisors}: a(n) = (1/2)*(A048146(n) + A162296(n)).
%H A325974 Antti Karttunen, <a href="/A325974/b325974.txt">Table of n, a(n) for n = 1..16384</a>
%H A325974 Antti Karttunen, <a href="/A325974/a325974.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%F A325974 a(n) = (1/2)*(A048146(n) + A162296(n)).
%F A325974 a(n) = A000203(n) - A325973(n).
%F A325974 a(n) = n - A325978(n).
%F A325974 Sum_{k=1..n} a(k) ~ c * n^2, where c = zeta(2)*(1/2 - 1/(4*zeta(3))) - 1/4 = 0.2303588390... . - _Amiram Eldar_, Feb 22 2024
%e A325974 For n = 36, its divisors are 1, 2, 3, 4, 6, 9, 12, 18, 36. Of these, non-unitary divisors are 2, 3, 6, 12 and 18 so A048146(36) = 2+3+6+12+18 = 41, while the nonsquarefree divisors are 4, 9, 12, 18 and 36, so A162296(36) = 4+9+12+18+36 = 79, thus a(36) = (41+79)/2 = 60.
%o A325974 (PARI)
%o A325974 A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
%o A325974 A048146(n) = (sigma(n)-A034448(n));
%o A325974 A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
%o A325974 A325974(n) = ((A048146(n)+A162296(n))/2);
%Y A325974 Cf. A000203, A005117, A034448, A048146, A162296, A325973, A325975, A325978.
%Y A325974 Cf. A002117, A013661.
%K A325974 nonn
%O A325974 1,4
%A A325974 _Antti Karttunen_, Jun 02 2019