This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A378699 #7 Dec 06 2024 11:10:11
%S A378699 7,2,1,0,3,1,0,0,0,2,1,2,0,0,0,0,0,2,1,0,0,1,0,1,0,1,0,0,1,0,0,0,0,0,
%T A378699 0,1,1,0,0,0,0,1,0,0,1,0,0,0,1,3,0,1,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,
%U A378699 1,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,1,1,0
%N A378699 Number of proper prime powers between powerful numbers that are not prime powers.
%C A378699 Within the sequence S = A001694 of powerful numbers, we have either proper prime powers k (in A246547) and numbers m that are not prime powers (in A286708). This sequence is the number of k between m.
%H A378699 Michael De Vlieger, <a href="/A378699/b378699.txt">Table of n, a(n) for n = 1..10000</a>
%e A378699 We partition S = A001694 by numbers m in A286708 (in brackets) and derive the following irregular table:
%e A378699 4, 8, 9, 16, 25, 27, 32, [36]; hence a(1) = 7,
%e A378699 49, 64, [72]; a(2) = 2,
%e A378699 81, [100]; a(3) = 1,
%e A378699 [108]; a(4) = 0,
%e A378699 121, 125, 128, [144]; a(5) = 3,
%e A378699 169, [196]; a(6) = 1,
%e A378699 [200]; a(7) = 0,
%e A378699 [216]; a(8) = 0,
%e A378699 [225]; a(9) = 0,
%e A378699 243, 256, [288]; a(10) = 2, etc.
%t A378699 nn = 2^16; s = Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}]; -1 + Length /@ TakeList[s, Differences@ Rest@ Position[s, _?(! PrimePowerQ[#] &) ][[All, 1]] ]
%Y A378699 Cf. A001694, A246547, A286708.
%K A378699 nonn,easy
%O A378699 1,1
%A A378699 _Michael De Vlieger_, Dec 04 2024