cp's OEIS Frontend

A377432 Number of perfect-powers x in the range prime(n) < x < prime(n+1).

%I A377432 #15 Nov 06 2024 04:35:08
%S A377432 0,1,0,2,0,1,0,0,2,0,2,0,0,0,1,0,0,1,0,0,0,1,0,0,1,0,0,0,0,2,1,0,0,1,
%T A377432 0,0,0,0,1,0,0,0,0,1,0,0,1,1,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,1,0,1,
%U A377432 0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0
%N A377432 Number of perfect-powers x in the range prime(n) < x < prime(n+1).
%C A377432 Perfect-powers (A001597) are numbers with a proper integer root, complement A007916.
%F A377432 a(n) + A377433(n) = A046933(n) = prime(n+1) - prime(n) - 1.
%e A377432 Between prime(4) = 7 and prime(5) = 11 we have perfect-powers 8 and 9, so a(4) = 2.
%t A377432 perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
%t A377432 Table[Length[Select[Range[Prime[n]+1, Prime[n+1]-1],perpowQ]],{n,100}]
%Y A377432 For prime-powers instead of perfect-powers we have A080101.
%Y A377432 Non-perfect-powers in the same range are counted by A377433.
%Y A377432 Positions of 1 are A377434.
%Y A377432 Positions of 0 are A377436.
%Y A377432 Positions of terms > 1 are A377466.
%Y A377432 For powers of 2 instead of primes we have A377467, for prime-powers A244508.
%Y A377432 A000040 lists the primes, differences A001223.
%Y A377432 A000961 lists the powers of primes, differences A057820.
%Y A377432 A001597 lists the perfect-powers, differences A053289.
%Y A377432 A007916 lists the non-perfect-powers, differences A375706.
%Y A377432 A046933 counts the interval from A008864(n) to A006093(n+1).
%Y A377432 A081676 gives the greatest perfect-power <= n.
%Y A377432 A246655 lists the prime-powers not including 1, complement A361102.
%Y A377432 A366833 counts prime-powers between primes, see A053706, A053607, A304521, A377286.
%Y A377432 A377468 gives the least perfect-power > n.
%Y A377432 Cf. A000015, A002808, A024619, A031218, A053707, A064113, A065514, A065890, A080769, A377051, A377282.
%K A377432 nonn
%O A377432 1,4
%A A377432 _Gus Wiseman_, Oct 31 2024