cp's OEIS Frontend

A377283 Nonnegative integers k such that either k = 0 or there is a perfect power x in the range prime(k) < x < prime(k+1).

%I A377283 #10 Dec 05 2024 21:47:49
%S A377283 0,2,4,6,9,11,15,18,22,25,30,31,34,39,44,47,48,53,54,61,66,68,72,78,
%T A377283 85,92,97,99,105,114,122,129,137,146,154,162,168,172,181,191,200,210,
%U A377283 217,219,228,240,251,263,269,274,283,295,306,309,319,327,329,342,357
%N A377283 Nonnegative integers k such that either k = 0 or there is a perfect power x in the range prime(k) < x < prime(k+1).
%C A377283 Perfect powers (A001597) are 1 and numbers with a proper integer root, complement A007916.
%e A377283 The first number-line below shows the perfect powers. The second shows each positive integer k at position prime(k).
%e A377283 -1-----4-------8-9------------16----------------25--27--------32------36----
%e A377283 ===1=2===3===4=======5===6=======7===8=======9==========10==11==========12==
%t A377283 perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
%t A377283 Select[Range[0,100],#==0||Length[Select[Range[Prime[#]+1,Prime[#+1]-1],perpowQ]]>0&]
%Y A377283 A version for prime powers is A377057, exclusive A377287.
%Y A377283 A version for squarefree numbers is A377431.
%Y A377283 Positions of positive terms in A377432 (counts perfect powers between primes).
%Y A377283 The case of a unique choice is A377434 (a subset).
%Y A377283 The complement (no choices) is A377436.
%Y A377283 The case of at least two choices is A377466 (a subset).
%Y A377283 Positions of last appearances in A378249.
%Y A377283 First-differences are A378251.
%Y A377283 This is A378365 - 1, union of A378356 - 1.
%Y A377283 A000040 lists the primes, differences A001223.
%Y A377283 A000961 lists the powers of primes, differences A057820.
%Y A377283 A001597 lists the perfect powers, differences A053289.
%Y A377283 A007916 lists the non perfect powers, differences A375706.
%Y A377283 A069623 counts perfect powers <= n.
%Y A377283 A076411 counts perfect powers < n.
%Y A377283 A131605 lists perfect powers that are not prime powers.
%Y A377283 Cf. A000015, A045542, A065514, A076412, A081676, A188951, A216765, A345531, A377468, A378035, A378250.
%K A377283 nonn
%O A377283 1,2
%A A377283 _Gus Wiseman_, Nov 21 2024