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A348030 a(n) = A003968(n) - n, where A003968 is multiplicative with a(p^e) = p*(p+1)^(e-1).

%I A348030 #15 May 29 2025 06:11:00
%S A348030 0,0,0,2,0,0,0,10,3,0,0,6,0,0,0,38,0,6,0,10,0,0,0,30,5,0,21,14,0,0,0,
%T A348030 130,0,0,0,36,0,0,0,50,0,0,0,22,15,0,0,114,7,10,0,26,0,42,0,70,0,0,0,
%U A348030 30,0,0,21,422,0,0,0,34,0,0,0,144,0,0,15,38,0,0,0,190,111,0,0,42,0,0,0,110,0,30,0,46
%N A348030 a(n) = A003968(n) - n, where A003968 is multiplicative with a(p^e) = p*(p+1)^(e-1).
%C A348030 Möbius transform of A348029(n), which is A003959(n) - sigma(n).
%H A348030 Antti Karttunen, <a href="/A348030/b348030.txt">Table of n, a(n) for n = 1..20000</a>
%F A348030 a(n) = A003968(n) - n.
%F A348030 a(n) = Sum_{d|n} A008683(n/d) * A348029(d).
%F A348030 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 + 1/(p^3 - p^2 - p)) - 1 = A104141/A005596 - 1 = 0.625665... . - _Amiram Eldar_, May 29 2025
%t A348030 f[p_, e_] := p*(p + 1)^(e - 1); a[n_] := Times @@ f @@@ FactorInteger[n] - n; Array[a, 100] (* _Amiram Eldar_, Oct 20 2021 *)
%o A348030 (PARI)
%o A348030 A003968(n) = {my(f=factor(n)); for (i=1, #f~, p= f[i, 1]; f[i, 1] = p*(p+1)^(f[i, 2]-1); f[i, 2] = 1); factorback(f); }
%o A348030 A348030(n) = (A003968(n)-n);
%Y A348030 Cf. A003959, A003968, A005117 (positions of zeros), A005596, A008683, A104141, A348029, A348036.
%K A348030 nonn
%O A348030 1,4
%A A348030 _Antti Karttunen_, Oct 19 2021