A349170 a(n) = Sum_{d|n} d * A003959(n/d), where A003959 is fully multiplicative with a(p) = (p+1).

1, 5, 7, 19, 11, 35, 15, 65, 37, 55, 23, 133, 27, 75, 77, 211, 35, 185, 39, 209, 105, 115, 47, 455, 91, 135, 175, 285, 59, 385, 63, 665, 161, 175, 165, 703, 75, 195, 189, 715, 83, 525, 87, 437, 407, 235, 95, 1477, 169, 455, 245, 513, 107, 875, 253, 975, 273, 295, 119, 1463, 123, 315, 555, 2059, 297, 805, 135, 665

Offset

1,2

Comments

Dirichlet convolution of A003959 with the identity function, A000027.

Dirichlet convolution of sigma (A000203) with A003968.

Links

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

Formula

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

a(n) = Sum_{d|n} A349171(d).

a(n) = Sum_{d|n} A000203(d) * A003968(n/d).

a(n) = A038040(n) + A349140(n).

For all n >= 1, a(n) >= A349129(n) >= A349130(n).

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

Mathematica

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

Prog

(PARI)
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A349170(n) = sumdiv(n, d, d*A003959(n/d));

Crossrefs

Cf. A000027, A000203, A003959, A003968, A038040, A349129, A349130, A349140, A349171 (Möbius transform), A349172, A349173.

Sequence in context: A028319 A116640 A116623 * A046151 A046078 A075409

Adjacent sequences: A349167 A349168 A349169 * A349171 A349172 A349173

Keyword

nonn,mult

Author

Antti Karttunen, Nov 09 2021

Status

approved