A349139 a(n) = Sum_{d|n} A322582(d) * A348507(n/d), where A322582(n) = n - A003958(n) and A348507(n) = A003959(n) - n.

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, 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, 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

Offset: 1

Author

Antti Karttunen, Nov 08 2021

Keywords

• nonn

Comments

Dirichlet convolution of A322582 with A348507.

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

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384

Crossrefs

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

Programs

Mathematica

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 *)

PARI

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A322582(n) = (n-A003958(n));
A348507(n) = (A003959(n)-n);
A349139(n) = sumdiv(n,d,A322582(d)*A348507(n/d));

Formula

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