A327527 Number of uniform divisors of n.

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 5, 2, 4, 4, 5, 2, 5, 2, 5, 4, 4, 2, 6, 3, 4, 4, 5, 2, 8, 2, 6, 4, 4, 4, 7, 2, 4, 4, 6, 2, 8, 2, 5, 5, 4, 2, 7, 3, 5, 4, 5, 2, 6, 4, 6, 4, 4, 2, 9, 2, 4, 5, 7, 4, 8, 2, 5, 4, 8, 2, 8, 2, 4, 5, 5, 4, 8, 2, 7, 5, 4, 2, 9, 4, 4, 4, 6, 2, 9, 4, 5, 4, 4, 4, 8, 2, 5, 5, 7, 2, 8, 2, 6, 8

Offset

1,2

Comments

A number is uniform if its prime multiplicities are all equal, meaning it is a power of a squarefree number. Uniform numbers are listed in A072774. The maximum uniform divisor of n is A327526(n).

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Gus Wiseman, Sequences counting and encoding certain classes of multisets.

Formula

From Amiram Eldar, Dec 19 2023: (Start)

a(n) = A034444(n) + A368251(n).

Sum_{k=1..n} a(k) ~ (n/zeta(2)) * (log(n) + 2*gamma - 1 - 2*zeta'(2)/zeta(2) + c * zeta(2)), where gamma is Euler's constant (A001620) and c = A368250. (End)

Example

The uniform divisors of 40 are {1, 2, 4, 5, 8, 10}, so a(40) = 6.

Mathematica

Table[Length[Select[Divisors[n], SameQ@@Last/@FactorInteger[#]&]], {n, 100}]
a[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, 1 + Total[2^Accumulate[Count[e, #] & /@ Range[Max[e], 1, -1]] - 1]]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Dec 19 2023 *)

Prog

(PARI)
isA072774(n) = { ispower(n, , &n); issquarefree(n); }; \\ From A072774
A327527(n) = sumdiv(n, d, isA072774(d)); \\ Antti Karttunen, Nov 13 2021

Crossrefs

See link for additional cross-references.

Cf. A000005, A000961, A005117, A006530, A007947, A071625, A112798.

Cf. A072774, A327526.

Cf. A001620, A013661, A306016, A368250, A368251.

Sequence in context: A301855 A080256 A351394 * A337454 A289849 A299701

Adjacent sequences: A327524 A327525 A327526 * A327528 A327529 A327530

Keyword

nonn

Author

Gus Wiseman, Sep 17 2019

Extensions

Data section extended up to 105 terms by Antti Karttunen, Nov 13 2021

Status

approved