%I #12 Nov 14 2021 17:44:43

%S 0,0,0,1,0,2,0,8,1,2,0,18,0,2,2,41,0,22,0,22,2,2,0,98,1,2,12,26,0,40,

%T 0,172,2,2,2,148,0,2,2,130,0,48,0,34,28,2,0,426,1,34,2,38,0,158,2,162,

%U 2,2,0,278,0,2,32,645,2,64,0,46,2,56,0,706,0,2,36,50,2,72,0,590,91,2,0,350,2,2,2,226,0,348

%N a(n) = Sum_{d|n} A322582(d) * A348507(n/d), where A322582(n) = n - A003958(n) and A348507(n) = A003959(n) - n.

%C Dirichlet convolution of A322582 with A348507.

%C Question: Is a(n) >= A305809(n) for all n?

%H Antti Karttunen, <a href="/A349139/b349139.txt">Table of n, a(n) for n = 1..16384</a>

%F a(n) = Sum_{d|n} A322582(d) * A348507(n/d).

%t f1[p_, e_] := (p - 1)^e; s1[1] = 0; s1[n_] := n - Times @@ f1 @@@ FactorInteger[n]; f2[p_, e_] := (p + 1)^e; s2[1] = 0; s2[n_] := Times @@ f2 @@@ FactorInteger[n] - n; a[n_] := DivisorSum[n, s1[#]*s2[n/#] &]; Array[a, 100] (* _Amiram Eldar_, Nov 08 2021 *)

%o (PARI)

%o A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };

%o A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };

%o A322582(n) = (n-A003958(n));

%o A348507(n) = (A003959(n)-n);

%o A349139(n) = sumdiv(n,d,A322582(d)*A348507(n/d));

%Y Cf. A003958, A003959, A305809, A322582, A348507, A349129.

%K nonn

%O 1,6

%A _Antti Karttunen_, Nov 08 2021