A394561 - OEIS

%I #8 Apr 08 2026 19:27:21

%S 0,1,0,1,1,0,1,2,0,2,1,1,1,3,0,0,1,2,3,1,0,1,2,5,0,1,2,2,0,1,3,0,0,4,

%T 1,1,1,0,0,2,2,5,0,3,1,5,3,1,0,3,1,1,7,0,3,1,2,1,2,0,1,2,3,0,1,0,2,7,

%U 3,3,0,2,7,1,5,0,1,0,3,1,4,1,0,0,3,7,3

%N Number of divisors d | A286708(n) such that d is an Achilles number (in A052486).

%C Cardinality of d in row A286708(n) of A027750 such that d is in A052486, where A286708 = A001694 \ A246547 \ {1}; i.e., powerful numbers without 1 and prime powers.

%H Michael De Vlieger, <a href="/A394561/b394561.txt">Table of n, a(n) for n = 1..10000</a>

%F Let s = A286708.

%F a(n) >= 1 for s(n) in A052486.

%F a(n) < A376514(s(n)).

%e Table of n, a(n) for select n:

%e   n a(n)   s(n)  List of d | s(n) such that d is in A052486

%e -----------------------------------------------------------

%e   1   0     36   {}

%e   2   1     72   {72}

%e   3   0    100   {}

%e   4   1    108   {108}

%e   5   1    144   {72}

%e   8   2    216   {72, 108}

%e  14   3    432   {72, 108, 432}

%e  24   5    864   {72, 108, 288, 432, 864}

%e  34   4   1296   {72, 108, 432, 648}

%e  53   7   2592   {72, 108, 288, 432, 648, 864, 2592}

%e 104   8   6912   {72, 108, 288, 432, 864, 1152, 3456, 6912}

%e 106   6   7200   {72, 200, 288, 800, 1800, 7200}

%t achillesQ[x_] := And[Min[#] > 1, GCD @@ # == 1] &[FactorInteger[x][[;; , -1]] ]; s = With[{nn = 2^13}, Rest@ Select[Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3] } ], Not @* PrimePowerQ] ]; Table[DivisorSum[s[[n]], 1 &, achillesQ], {n, 100}]

%Y Cf. A001694, A027750, A052486, A286708, A376514.

%K nonn

%O 1,8

%A _Michael De Vlieger_, Mar 30 2026