A349129 a(n) = Sum_{d|n} A003958(d) * A003959(n/d), where A003958 is fully multiplicative with a(p) = (p-1), and A003959 is fully multiplicative with a(p) = (p+1).

1, 4, 6, 13, 10, 24, 14, 40, 28, 40, 22, 78, 26, 56, 60, 121, 34, 112, 38, 130, 84, 88, 46, 240, 76, 104, 120, 182, 58, 240, 62, 364, 132, 136, 140, 364, 74, 152, 156, 400, 82, 336, 86, 286, 280, 184, 94, 726, 148, 304, 204, 338, 106, 480, 220, 560, 228, 232, 118, 780, 122, 248, 392, 1093, 260, 528, 134, 442, 276

Offset: 1

Author

Antti Karttunen, Nov 09 2021

Keywords

• nonn
• mult

Comments

Dirichlet convolution of A003958 with A003959.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A003958, A003959, A168066, A349139.

Cf. also A349130, A349131, A349132, A349133 and A349170, A349171, A349172, A349173.

Programs

Mathematica

f[p_, e_] := ((p + 1)^(e + 1) - (p - 1)^(e + 1))/2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 09 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); };
A349129(n) = sumdiv(n,d,A003958(d)*A003959(n/d));

Formula

Multiplicative with a(p^e) = ((p+1)^(e+1) - (p-1)^(e+1))/2. - Amiram Eldar, Nov 09 2021

For all n >= 1, A349130(n) <= a(n) <= A349170(n).