A349143 a(n) = Sum_{d|n} A038040(d) * A348507(n/d), where A038040(n) = n*tau(n), A348507(n) = A003959(n) - n, and A003959 is fully multiplicative with a(p) = (p+1).

0, 1, 1, 9, 1, 16, 1, 51, 13, 22, 1, 114, 1, 28, 25, 233, 1, 145, 1, 168, 31, 40, 1, 590, 21, 46, 106, 222, 1, 310, 1, 939, 43, 58, 37, 915, 1, 64, 49, 896, 1, 406, 1, 330, 262, 76, 1, 2570, 29, 297, 61, 384, 1, 1012, 49, 1202, 67, 94, 1, 2040, 1, 100, 340, 3489, 55, 598, 1, 492, 79, 574, 1, 4457, 1, 118, 360, 546, 55

Offset

1,4

Comments

Dirichlet convolution of A348507 with A038040, which is the Dirichlet convolution of the identity function (A000027) with itself.

Dirichlet convolution of the identity function (A000027) with A349140.

Dirichlet convolution of sigma (A000203) with A349141.

Dirichlet convolution of A060640 with A348971.

Links

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

Formula

a(n) = Sum_{d|n} A038040(n/d) * A348507(d).

a(n) = Sum_{d|n} d * A349140(n/d).

a(n) = Sum_{d|n} A000203(d) * A349141(n/d).

a(n) = Sum_{d|n} A060640(d) * A348971(n/d).

For all n >= 1, a(n) >= A349123(n) >= A348983(n).

Mathematica

f[p_, e_] := (p + 1)^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := DivisorSum[n, #*DivisorSigma[0, #]*(s[n/#] - n/#) &]; Array[a, 100] (* Amiram Eldar, Nov 08 2021 *)

Prog

(PARI)
A038040(n) = (n*numdiv(n));
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348507(n) = (A003959(n) - n);
A349143(n) = sumdiv(n, d, A038040(d)*A348507(n/d));

Crossrefs

Cf. A000005, A000027, A000203, A003959, A038040, A060640, A348507, A348971, A349140, A349141, A349142.

Cf. also A349123, A348983.

Sequence in context: A061215 A072448 A145966 * A014721 A054018 A342637

Adjacent sequences: A349140 A349141 A349142 * A349144 A349145 A349146

Keyword

nonn

Author

Antti Karttunen, Nov 08 2021

Status

approved