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

1,2

Comments

Dirichlet convolution of A001615 with A003959.

Links

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

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

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 *)

Prog

(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));

Crossrefs

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

Sequence in context: A085796 A280641 A005887 * A119875 A053189 A156231

Adjacent sequences: A349169 A349170 A349171 * A349173 A349174 A349175

Keyword

nonn,mult

Author

Antti Karttunen, Nov 09 2021

Status

approved