A378899 - OEIS

%I #6 Dec 12 2024 15:24:54

%S 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 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 13,7,4,7,5,4,13,2,6,10,11,29,4,2,11,1,28,2,14

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

%H Michael De Vlieger, <a href="/A378899/b378899.txt">Table of n, a(n) for n = 0..10000</a>

%F a(0) = pi(36) = A000720(36) = 11.

%F For n > 0, a(n) = pi(A286708(n+1)) - pi(A286708(n)).

%e Let s = A286708.

%e a(0) = 11 since {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31} are primes less than s(1) = 36.

%e 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 a(2) = 5 since {73, 79, 83, 89, 97} are primes p such that s(2) < p < s(3), where s(3) = 100.

%e a(3) = 3 since {101, 103, 107} are primes p such that s(3) < p < s(4), where s(4) = 108, etc.

%t s = With[{nn = 5000},

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

%t   Not@*PrimePowerQ]];

%t {PrimePi[s[[1]]]}~Join~Differences@ Map[PrimePi, s]

%Y Cf. A000040, A000720, A240590, A286708, A378699.

%K nonn,easy,new

%O 0,1

%A _Michael De Vlieger_, Dec 10 2024