A378178 - OEIS

%I #11 Nov 27 2024 18:34:56

%S 0,0,0,0,0,0,1,0,1,2,0,0,1,4,0,1,1,0,6,0,1,3,2,2,3,7,1,5,3,4,1,0,0,2,

%T 6,4,7,2,5,3,6,1,3,11,2,2,0,10,2,13,6,3,7,2,3,5,14,7,2,1,1,3,11,2,2,

%U 17,3,8,1,11,9,2,6,0,11,1,0,20,5,4,4,24,23

%N Number of powerful k between consecutive perfect (or proper) prime powers.

%C Powerful k between perfect prime powers are in A286708.

%H Michael De Vlieger, <a href="/A378178/b378178.txt">Table of n, a(n) for n = 1..10000</a>

%e Let s = A001694 and t = A246547.

%e a(1..6) = 0 since s(9) = 36 is the smallest powerful number that is not a prime power.

%e a(7) = 1, since only s(9) = 36 is such that t(8) < 36 < t(9), i.e., 32 < 36 < 49.

%e a(8) = 0, since there are no powerful numbers between t(9) = 49 and t(10) = 64.

%e a(9) = 1, since only s(12) = 72 is such that t(10) < 72 < t(9), i.e., 64 < 72 < 81.

%e a(10) = 2, since t(11) < s(14..15) < t(12), i.e., 100 and 108 exceed 81 but not 121, etc.

%t nn = 2^16;

%t s = Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}];

%t -1 + Length /@ TakeList[s, Differences@ Position[s, _?PrimePowerQ][[All, 1]] ]

%o (PARI) lista(nn) = my(v=select(x->(isprimepower(x) > 1), [1..nn])); vector(#v-1, k, #select(ispowerful, [v[k]+1..v[k+1]-1])); \\ _Michel Marcus_, Nov 25 2024

%Y Cf. A001694, A246547, A286708.

%K nonn,easy,new

%O 1,10

%A _Michael De Vlieger_, Nov 24 2024