A382872 For n >= 1, a(n) is the number of divisors of the Pillai's arithmetical function: Sum_{k=1..n} gcd(k, n) (A018804).

1, 2, 2, 4, 3, 4, 2, 6, 4, 4, 4, 8, 3, 4, 6, 10, 4, 6, 2, 12, 4, 6, 6, 9, 4, 6, 5, 8, 4, 8, 2, 10, 8, 6, 6, 16, 2, 4, 4, 18, 5, 8, 4, 16, 8, 8, 4, 20, 4, 8, 8, 12, 8, 6, 8, 12, 4, 6, 6, 24, 3, 4, 8, 9, 9, 12, 4, 16, 9, 8, 4, 24, 4, 4, 6, 8, 8, 8, 2, 20

Offset

1,2

Comments

a(n) is from A005408 for n from {1, 5, 13, 24, 27, 41, 61, 64, 65, 69, 99, 113, ...}.

a(n) is from A065091 for n from {5, 13, 27, 41, 61, 135, 181, 205, 313, 421, ...}.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

a(n) = A000005(A018804(n)).

a(A005382(n)) = 2.

a(A067756(n)) = 3.

a(A277201(n)) = 5.

Example

For n = 5, a(5) = A000005(A018804(5)) = A000005(9) = 3.

Maple

f:= proc(n) local i; numtheory:-tau(add(igcd(i, n), i=1..n)) end proc:
map(f, [$1..100]); # Robert Israel, May 07 2025

Mathematica

f[p_, e_] := (e*(p - 1)/p + 1)*p^e; a[n_] := DivisorSigma[0, Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Apr 07 2025 *)

Prog

(PARI) a(n) = numdiv(sumdiv(n, d, n*eulerphi(d)/d)); \\ Michel Marcus, Apr 07 2025

Crossrefs

Cf. A000005, A005382, A005408, A018804, A065091, A067756, A277201.

Sequence in context: A371409 A338756 A078317 * A105016 A377369 A074747

Adjacent sequences: A382869 A382870 A382871 * A382873 A382874 A382875

Keyword

nonn

Author

Ctibor O. Zizka, Apr 07 2025

Status

approved