A349142 a(n) = Sum_{d|n} psi(n/d) * A348507(d), where psi is Dedekind psi (A001615), A348507(n) = A003959(n) - n, and A003959 is fully multiplicative with a(p) = (p+1).

0, 1, 1, 8, 1, 13, 1, 40, 11, 17, 1, 80, 1, 21, 19, 164, 1, 99, 1, 112, 23, 29, 1, 364, 17, 33, 77, 144, 1, 191, 1, 604, 31, 41, 27, 528, 1, 45, 35, 524, 1, 243, 1, 208, 165, 53, 1, 1424, 23, 187, 43, 240, 1, 597, 35, 684, 47, 65, 1, 1072, 1, 69, 209, 2084, 39, 347, 1, 304, 55, 327, 1, 2244, 1, 81, 221, 336, 39, 399

Offset: 1

Author

Antti Karttunen, Nov 08 2021

Keywords

• nonn

Comments

Dirichlet convolution of A001615 with A348507.

Links

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

Crossrefs

Cf. A001615, A003959, A327251, A348507, A349140, A349141, A349143, A349172.

Cf. also A347132, A348982.

Programs

Mathematica

f1[p_, e_] := (p + 1)*p^(e - 1); psi[1] = 1; psi[n_] := Times @@ f1 @@@ FactorInteger[n]; f2[p_, e_] := (p + 1)^e; s[1] = 1; s[n_] := Times @@ f2 @@@ FactorInteger[n]; a[n_] := DivisorSum[n, (s[#] - #)*psi[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 08 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))); \\ After code in A001615
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348507(n) = (A003959(n) - n);
A349142(n) = sumdiv(n,d,A001615(d)*A348507(n/d));

Formula

a(n) = Sum_{d|n} A001615(n/d) * A348507(d).

For all n >= 1, a(n) >= A347132(n) >= A348982(n).

a(n) = A349172(n) - A327251(n). - Antti Karttunen, Nov 14 2021