A349172 a(n) = Sum_{d|n} psi(d) * A003959(n/d), where A003959 is fully multiplicative with a(p) = (p+1), and psi is Dedekind psi function, A001615.

1, 6, 8, 24, 12, 48, 16, 84, 44, 72, 24, 192, 28, 96, 96, 276, 36, 264, 40, 288, 128, 144, 48, 672, 102, 168, 212, 384, 60, 576, 64, 876, 192, 216, 192, 1056, 76, 240, 224, 1008, 84, 768, 88, 576, 528, 288, 96, 2208, 184, 612, 288, 672, 108, 1272, 288, 1344, 320, 360, 120, 2304, 124, 384, 704, 2724, 336, 1152, 136

Offset: 1

Author

Antti Karttunen, Nov 09 2021

Keywords

• nonn
• mult

Comments

Dirichlet convolution of A001615 with A003959.

Links

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

Crossrefs

Cf. A001615, A003959, A327251, A349142, A349170, A349171, A349172, A349173.

Programs

Mathematica

f[p_, e_] := (p + 2)*(p + 1)^e - (p + 1)*p^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 09 2021 *)

PARI

A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1)));
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A349172(n) = sumdiv(n,d,A001615(d)*A003959(n/d));

Formula

a(n) = Sum_{d|n} A001615(d) * A003959(n/d).

a(n) = A327251(n) + A349142(n).

For all n >= 1, a(n) >= A349132(n).

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