A377283 - OEIS

%I #10 Dec 05 2024 21:47:49

%S 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 85,92,97,99,105,114,122,129,137,146,154,162,168,172,181,191,200,210,

%U 217,219,228,240,251,263,269,274,283,295,306,309,319,327,329,342,357

%N 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 Perfect powers (A001597) are 1 and numbers with a proper integer root, complement A007916.

%e The first number-line below shows the perfect powers. The second shows each positive integer k at position prime(k).

%e -1-----4-------8-9------------16----------------25--27--------32------36----

%e ===1=2===3===4=======5===6=======7===8=======9==========10==11==========12==

%t perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;

%t Select[Range[0,100],#==0||Length[Select[Range[Prime[#]+1,Prime[#+1]-1],perpowQ]]>0&]

%Y A version for prime powers is A377057, exclusive A377287.

%Y A version for squarefree numbers is A377431.

%Y Positions of positive terms in A377432 (counts perfect powers between primes).

%Y The case of a unique choice is A377434 (a subset).

%Y The complement (no choices) is A377436.

%Y The case of at least two choices is A377466 (a subset).

%Y Positions of last appearances in A378249.

%Y First-differences are A378251.

%Y This is A378365 - 1, union of A378356 - 1.

%Y A000040 lists the primes, differences A001223.

%Y A000961 lists the powers of primes, differences A057820.

%Y A001597 lists the perfect powers, differences A053289.

%Y A007916 lists the non perfect powers, differences A375706.

%Y A069623 counts perfect powers <= n.

%Y A076411 counts perfect powers < n.

%Y A131605 lists perfect powers that are not prime powers.

%Y Cf. A000015, A045542, A065514, A076412, A081676, A188951, A216765, A345531, A377468, A378035, A378250.

%K nonn

%O 1,2

%A _Gus Wiseman_, Nov 21 2024