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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).

%I A382872 #19 May 21 2025 00:52:14
%S A382872 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,
%T A382872 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,
%U A382872 9,12,4,16,9,8,4,24,4,4,6,8,8,8,2,20
%N 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).
%C A382872 a(n) is from A005408 for n from {1, 5, 13, 24, 27, 41, 61, 64, 65, 69, 99, 113, ...}.
%C A382872 a(n) is from A065091 for n from {5, 13, 27, 41, 61, 135, 181, 205, 313, 421, ...}.
%H A382872 Robert Israel, <a href="/A382872/b382872.txt">Table of n, a(n) for n = 1..10000</a>
%F A382872 a(n) = A000005(A018804(n)).
%F A382872 a(A005382(n)) = 2.
%F A382872 a(A067756(n)) = 3.
%F A382872 a(A277201(n)) = 5.
%e A382872 For n = 5, a(5) = A000005(A018804(5)) = A000005(9) = 3.
%p A382872 f:= proc(n) local i; numtheory:-tau(add(igcd(i,n),i=1..n)) end proc:
%p A382872 map(f, [$1..100]); # _Robert Israel_, May 07 2025
%t A382872 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 *)
%o A382872 (PARI) a(n) = numdiv(sumdiv(n, d, n*eulerphi(d)/d)); \\ _Michel Marcus_, Apr 07 2025
%Y A382872 Cf. A000005, A005382, A005408, A018804, A065091, A067756, A277201.
%K A382872 nonn
%O A382872 1,2
%A A382872 _Ctibor O. Zizka_, Apr 07 2025