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%I #24 Dec 04 2024 20:38:05
%S 2,0,2,3,0,2,4,3,4,8,0,1,8,14,1,7,7,4,25,2,15,15,17,16,10,45,2,44,20,
%T 26,18,0,2,28,52,36,42,32,45,45,47,19,30,106,36,35,4,114,28,135,89,42,
%U 87,42,34,66,192,106,56,23,39,37,165,49,37,262,58,160,22
%N Number of primes between consecutive terms of A246547 (prime powers p^k, k >= 2).
%C Does this sequence have any terms appearing infinitely often? In particular, are {2, 5, 11, 32, 77} the only zeros? As an example, {121, 122, 123, 124, 125} is an interval containing no primes, corresponding to a(11) = 0. - _Gus Wiseman_, Dec 02 2024
%H Michael De Vlieger, <a href="/A067871/b067871.txt">Table of n, a(n) for n = 1..10000</a> (first 667 terms from Lei Zhou)
%F a(n) = A000720(A025475(n+3)) - A000720(A025475(n+2)) - _David Wasserman_, Dec 20 2002
%e The first few prime powers A246547 are 4, 8, 9, 16. The first few primes are 2, 3, 5, 7, 11, 13. We have (4), 5, 7, (8), (9), 11, 13, (16) and so the sequence begins with 2, 0, 2.
%e The initial terms count the following sets of primes: {5,7}, {}, {11,13}, {17,19,23}, {}, {29,31}, {37,41,43,47}, ... - _Gus Wiseman_, Dec 02 2024
%t t = {}; cnt = 0; Do[If[PrimePowerQ[n], If[FactorInteger[n][[1, 2]] == 1, cnt++, AppendTo[t, cnt]; cnt = 0]], {n, 4 + 1, 30000}]; t (* _T. D. Noe_, May 21 2013 *)
%t nn = 2^20; Differences@ Map[PrimePi, Select[Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], PrimePowerQ] ] ] (* _Michael De Vlieger_, Oct 26 2023 *)
%Y For primes between nonsquarefree numbers we have A236575.
%Y For composite instead of prime we have A378456.
%Y A000015 gives the least prime power >= n.
%Y A000040 lists the primes, differences A001223.
%Y A000961 lists the powers of primes, differences A057820.
%Y A080101 counts prime powers between primes.
%Y A246547 lists the non prime prime powers, differences A053707.
%Y A246655 lists the prime powers not including 1, complement A361102.
%Y Cf. A001597, A024619, A031218, A046933, A276781, A345531, A366833, A377051, A377057, A377282, A377286-A377288.
%K nonn,easy
%O 1,1
%A _Jon Perry_, Mar 07 2002
%E More terms from _David Wasserman_, Dec 20 2002
%E Definition clarified by _N. J. A. Sloane_, Oct 27 2023