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%I #15 Dec 19 2023 09:19:55
%S 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,
%T 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,
%U 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
%N Number of uniform divisors of n.
%C 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).
%H Antti Karttunen, <a href="/A327527/b327527.txt">Table of n, a(n) for n = 1..65537</a>
%H Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vSX9dPMGJhxB8rOknCGvOs6PiyhupdWNpqLsnphdgU6MEVqFBnWugAXidDhwHeKqZe_YnUqYeGOXsOk/pub">Sequences counting and encoding certain classes of multisets</a>.
%F From _Amiram Eldar_, Dec 19 2023: (Start)
%F a(n) = A034444(n) + A368251(n).
%F 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)
%e The uniform divisors of 40 are {1, 2, 4, 5, 8, 10}, so a(40) = 6.
%t Table[Length[Select[Divisors[n],SameQ@@Last/@FactorInteger[#]&]],{n,100}]
%t 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 *)
%o (PARI)
%o isA072774(n) = { ispower(n, , &n); issquarefree(n); }; \\ From A072774
%o A327527(n) = sumdiv(n,d,isA072774(d)); \\ _Antti Karttunen_, Nov 13 2021
%Y See link for additional cross-references.
%Y Cf. A000005, A000961, A005117, A006530, A007947, A071625, A112798.
%Y Cf. A072774, A327526.
%Y Cf. A001620, A013661, A306016, A368250, A368251.
%K nonn
%O 1,2
%A _Gus Wiseman_, Sep 17 2019
%E Data section extended up to 105 terms by _Antti Karttunen_, Nov 13 2021
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