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A378899 Number of primes between successive powerful numbers k that are not prime powers (i.e., k in A286708).

%I A378899 #6 Dec 12 2024 15:24:54
%S A378899 11,9,5,3,6,10,2,1,1,13,5,11,1,5,2,7,3,10,13,4,0,15,2,11,4,9,1,4,13,7,
%T A378899 2,1,9,10,6,1,2,9,12,7,4,18,5,4,17,0,8,3,13,23,2,23,10,1,15,0,7,18,3,
%U A378899 13,7,4,7,5,4,13,2,6,10,11,29,4,2,11,1,28,2,14
%N A378899 Number of primes between successive powerful numbers k that are not prime powers (i.e., k in A286708).
%H A378899 Michael De Vlieger, <a href="/A378899/b378899.txt">Table of n, a(n) for n = 0..10000</a>
%F A378899 a(0) = pi(36) = A000720(36) = 11.
%F A378899 For n > 0, a(n) = pi(A286708(n+1)) - pi(A286708(n)).
%e A378899 Let s = A286708.
%e A378899 a(0) = 11 since {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31} are primes less than s(1) = 36.
%e A378899 a(1) = 9 since {37, 41, 43, 47, 53, 59, 61, 67, 71} are primes that exceed s(1) but not s(2) = 72.
%e A378899 a(2) = 5 since {73, 79, 83, 89, 97} are primes p such that s(2) < p < s(3), where s(3) = 100.
%e A378899 a(3) = 3 since {101, 103, 107} are primes p such that s(3) < p < s(4), where s(4) = 108, etc.
%t A378899 s = With[{nn = 5000},
%t A378899   Select[Rest@ Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}],
%t A378899   Not@*PrimePowerQ]];
%t A378899 {PrimePi[s[[1]]]}~Join~Differences@ Map[PrimePi, s]
%Y A378899 Cf. A000040, A000720, A240590, A286708, A378699.
%K A378899 nonn,easy
%O A378899 0,1
%A A378899 _Michael De Vlieger_, Dec 10 2024