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A348944 a(n) = (1/2) * (A003959(n)+A034448(n)), where A003959 is multiplicative with a(p^e) = (p+1)^e and A034448 (usigma) is multiplicative with a(p^e) = (p^e)+1.

%I A348944 #10 Feb 11 2022 11:23:35
%S A348944 1,3,4,7,6,12,8,18,13,18,12,28,14,24,24,49,18,39,20,42,32,36,24,72,31,
%T A348944 42,46,56,30,72,32,138,48,54,48,97,38,60,56,108,42,96,44,84,78,72,48,
%U A348944 196,57,93,72,98,54,138,72,144,80,90,60,168,62,96,104,397,84,144,68,126,96,144,72,261,74,114,124,140,96
%N A348944 a(n) = (1/2) * (A003959(n)+A034448(n)), where A003959 is multiplicative with a(p^e) = (p+1)^e and A034448 (usigma) is multiplicative with a(p^e) = (p^e)+1.
%C A348944 This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 36 = 2^2 * 3^2, where a(36) = 97 != 91 = 7*13 = a(4)*a(9).
%H A348944 Antti Karttunen, <a href="/A348944/b348944.txt">Table of n, a(n) for n = 1..20000</a>
%t A348944 f1[p_, e_] := (p + 1)^e; f2[p_, e_] := p^e + 1; a[1] = 1; a[n_] := (Times @@ f1 @@@ (f = FactorInteger[n]) + Times @@ f2 @@@ f) / 2; Array[a, 100] (* _Amiram Eldar_, Nov 05 2021 *)
%o A348944 (PARI)
%o A348944 A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
%o A348944 A034448(n) = { my(f = factor(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
%o A348944 A348944(n) = ((1/2)*(A003959(n)+A034448(n)));
%Y A348944 Arithmetic mean of A003959 and A034448.
%Y A348944 Cf. A348732, A348733, A348945.
%Y A348944 Cf. also A325973.
%K A348944 nonn
%O A348944 1,2
%A A348944 _Antti Karttunen_, Nov 05 2021