cp's OEIS Frontend

A377434 Numbers k such that there is a unique perfect-power x in the range prime(k) < x < prime(k+1).

%I A377434 #12 Nov 04 2024 09:15:43
%S A377434 2,6,15,18,22,25,31,34,39,44,47,48,53,54,61,66,68,72,78,85,92,97,99,
%T A377434 105,114,122,129,137,146,154,162,168,172,181,191,200,210,217,219,228,
%U A377434 240,251,263,269,274,283,295,306,309,319,329,342,357,367,378,393,400
%N A377434 Numbers k such that there is a unique perfect-power x in the range prime(k) < x < prime(k+1).
%C A377434 Perfect-powers (A001597) are numbers with a proper integer root, complement A007916.
%e A377434 Primes 4 and 5 are 7 and 11, and the interval (8,9,10) contains two perfect-powers (8,9), so 4 is not in the sequence.
%e A377434 Primes 5 and 6 are 11 and 13, and the interval (12) contains no perfect-powers, so 5 is not in the sequence.
%e A377434 Primes 6 and 7 are 13 and 17, and the interval (14,15,16) contains just one perfect-power (16), so 6 is in the sequence.
%t A377434 perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
%t A377434 Select[Range[100],Length[Select[Range[Prime[#]+1,Prime[#+1]-1],perpowQ]]==1&]
%Y A377434 For powers of 2 instead of primes see A013597, A014210, A014234, A244508, A377467.
%Y A377434 For prime-powers we have A377287.
%Y A377434 For squarefree numbers see A377430, A061398, A377431, A068360.
%Y A377434 These are the positions of 1 in A377432.
%Y A377434 For no perfect-powers we have A377436.
%Y A377434 For more than one perfect-power we have A377466.
%Y A377434 A000015 gives the least prime-power >= n.
%Y A377434 A000040 lists the primes, differences A001223.
%Y A377434 A000961 lists the powers of primes, differences A057820.
%Y A377434 A001597 lists the perfect-powers, differences A053289, seconds A376559.
%Y A377434 A007916 lists the non-perfect-powers, differences A375706, seconds A376562.
%Y A377434 A031218 gives the greatest prime-power <= n.
%Y A377434 A046933 counts the interval from A008864(n) to A006093(n+1).
%Y A377434 A065514 gives the greatest prime-power < prime(n), difference A377289.
%Y A377434 A081676 gives the greatest perfect-power <= n.
%Y A377434 A131605 lists perfect-powers that are not prime-powers.
%Y A377434 A345531 gives the least prime-power > prime(n), difference A377281.
%Y A377434 A366833 counts prime-powers between primes, see A053607, A304521.
%Y A377434 A377468 gives the least perfect-power > n.
%Y A377434 Cf. A006549, A023055, A045542, A052410, A069623, A080101, A216765, A224363, A246655, A336416, A375740, A376560, A376561, A377057.
%K A377434 nonn
%O A377434 1,1
%A A377434 _Gus Wiseman_, Nov 02 2024