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A348029 a(n) = A003959(n) - sigma(n), where A003959 is multiplicative with a(p^e) = (p+1)^e and sigma is the sum of divisors.

%I A348029 #14 May 29 2025 06:11:31
%S A348029 0,0,0,2,0,0,0,12,3,0,0,8,0,0,0,50,0,9,0,12,0,0,0,48,5,0,24,16,0,0,0,
%T A348029 180,0,0,0,53,0,0,0,72,0,0,0,24,18,0,0,200,7,15,0,28,0,72,0,96,0,0,0,
%U A348029 48,0,0,24,602,0,0,0,36,0,0,0,237,0,0,20,40,0,0,0,300,135,0,0,64,0,0,0,144,0,54,0,48,0
%N A348029 a(n) = A003959(n) - sigma(n), where A003959 is multiplicative with a(p^e) = (p+1)^e and sigma is the sum of divisors.
%C A348029 Inverse Möbius transform of A348030.
%H A348029 Antti Karttunen, <a href="/A348029/b348029.txt">Table of n, a(n) for n = 1..20000</a>
%F A348029 a(n) = A003959(n) - A000203(n).
%F A348029 a(n) = Sum_{d|n} A348030(d).
%F A348029 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 + 1/(p^2-p-1)) - Pi^2/6 = A065488 - A013661 = 1.0291786... . - _Amiram Eldar_, May 29 2025
%t A348029 f[p_, e_] := (p + 1)^e; a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - DivisorSigma[1, n]; Array[a, 100] (* _Amiram Eldar_, Oct 20 2021 *)
%o A348029 (PARI)
%o A348029 A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
%o A348029 A348029(n) = (A003959(n)-sigma(n));
%Y A348029 Cf. A000203, A003959, A005117 (positions of zeros), A013661, A065488, A348030.
%K A348029 nonn
%O A348029 1,4
%A A348029 _Antti Karttunen_, Oct 20 2021