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A378251 Number of primes between consecutive perfect powers, zeros omitted.

%I A378251 #22 Jan 25 2025 03:23:40
%S A378251 2,2,2,3,2,4,3,4,3,5,1,3,5,5,3,1,5,1,7,5,2,4,6,7,7,5,2,6,9,8,7,8,9,8,
%T A378251 8,6,4,9,10,9,10,7,2,9,12,11,12,6,5,9,12,11,3,10,8,2,13,15,10,11,15,7,
%U A378251 9,12,13,11,12,17,2,11,16,16,13,17,15,14,16,15
%N A378251 Number of primes between consecutive perfect powers, zeros omitted.
%C A378251 First differences of A377283 and A378365. Run-lengths of A378035 and A378249.
%C A378251 Perfect powers (A001597) are 1 and numbers with a proper integer root, complement A007916.
%H A378251 Robert Israel, <a href="/A378251/b378251.txt">Table of n, a(n) for n = 1..10000</a>
%e A378251 The first number line below shows the perfect powers. The second shows each prime. To get a(n) we count the primes between consecutive perfect powers, skipping the cases where there are none.
%e A378251 -1-----4-------8-9------------16----------------25--27--------32------36----
%e A378251 ===2=3===5===7======11==13======17==19======23==========29==31==========37==
%p A378251 N:= 10^6: # to use perfect powers up to N
%p A378251 PP:= {1,seq(seq(i^j,j=2..ilog[i](N)),i=2..isqrt(N))}:
%p A378251 PP:= sort(convert(PP,list)):
%p A378251 M:= map(numtheory:-pi, PP):
%p A378251 subs(0=NULL, M[2..-1]-M[1..-2]): # _Robert Israel_, Jan 23 2025
%t A378251 radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
%t A378251 Length/@Split[Table[NestWhile[#+1&,Prime[n],radQ[#]&],{n,100}]]
%Y A378251 Same as A080769 with 0's removed (which were at positions A274605).
%Y A378251 First differences of A377283 and A378365 (union of A378356).
%Y A378251 Run-lengths of A378035 (union A378253) and A378249 (union A378250).
%Y A378251 The version for nonprime prime powers is A378373, with zeros A067871.
%Y A378251 A000040 lists the primes, differences A001223.
%Y A378251 A000961 lists the powers of primes, differences A057820.
%Y A378251 A001597 lists the perfect powers, differences A053289, run-lengths of A377468.
%Y A378251 A007916 lists the non-perfect powers, differences A375706.
%Y A378251 A069623 counts perfect powers <= n.
%Y A378251 A076411 counts perfect powers < n.
%Y A378251 A131605 lists perfect powers that are not prime powers.
%Y A378251 A377432 counts perfect powers between primes, see A377434, A377436, A377466.
%Y A378251 Cf. A007918, A023055, A045542, A052410, A076412, A081676, A188951.
%Y A378251 Cf. A216765, A377057, A377431, A378355.
%K A378251 nonn,look
%O A378251 1,1
%A A378251 _Gus Wiseman_, Nov 23 2024