cp's OEIS Frontend

A377466 Numbers k such that there is more than one perfect power x in the range prime(k) < x < prime(k+1).

%I A377466 #26 Nov 06 2024 04:35:19
%S A377466 4,9,11,30,327,445,3512,7789,9361,26519413
%N A377466 Numbers k such that there is more than one perfect power x in the range prime(k) < x < prime(k+1).
%C A377466 Perfect powers (A001597) are numbers with a proper integer root, the complement of A007916.
%C A377466 Is this sequence finite?
%C A377466 The Redmond-Sun conjecture (see A308658) implies that this sequence is finite. - _Pontus von Brömssen_, Nov 05 2024
%F A377466 a(n) = A000720(A116086(n)) = A000720(A116455(n)) for n <= 10. This would hold for all n if there do not exist more than two perfect powers between any two consecutive primes, which is implied by the Redmond-Sun conjecture. - _Pontus von Brömssen_, Nov 05 2024
%e A377466 Primes 9 and 10 are 23 and 29, and the interval (24,25,26,27,28) contains two perfect powers (25,27), so 9 is in the sequence.
%t A377466 perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
%t A377466 Select[Range[100],Count[Range[Prime[#]+1, Prime[#+1]-1],_?perpowQ]>1&]
%o A377466 (Python)
%o A377466 from itertools import islice
%o A377466 from sympy import prime
%o A377466 from gmpy2 import is_power, next_prime
%o A377466 def A377466_gen(startvalue=1): # generator of terms >= startvalue
%o A377466     k = max(startvalue,1)
%o A377466     p = prime(k)
%o A377466     while (q:=next_prime(p)):
%o A377466         c = 0
%o A377466         for i in range(p+1,q):
%o A377466             if is_power(i):
%o A377466                 c += 1
%o A377466                 if c>1:
%o A377466                     yield k
%o A377466                     break
%o A377466         k += 1
%o A377466         p = q
%o A377466 A377466_list = list(islice(A377466_gen(),9)) # _Chai Wah Wu_, Nov 04 2024
%Y A377466 For powers of 2 see A013597, A014210, A014234, A188951, A244508, A377467.
%Y A377466 For no prime-powers we have A377286, ones in A080101.
%Y A377466 For a unique prime-power we have A377287.
%Y A377466 For squarefree numbers see A377430, A061398, A377431, A068360, A224363.
%Y A377466 These are the positions of terms > 1 in A377432.
%Y A377466 For a unique perfect power we have A377434.
%Y A377466 For no perfect powers we have A377436.
%Y A377466 A000015 gives the least prime power >= n.
%Y A377466 A000040 lists the primes, differences A001223.
%Y A377466 A000961 lists the powers of primes, differences A057820.
%Y A377466 A001597 lists the perfect powers, differences A053289, seconds A376559.
%Y A377466 A007916 lists the non-perfect-powers, differences A375706, seconds A376562.
%Y A377466 A046933 counts the interval from A008864(n) to A006093(n+1).
%Y A377466 A081676 gives the greatest perfect power <= n.
%Y A377466 A131605 lists perfect powers that are not prime-powers.
%Y A377466 A246655 lists the prime-powers not including 1, complement A361102.
%Y A377466 A366833 counts prime-powers between primes, see A053607, A304521.
%Y A377466 A377468 gives the least perfect power > n.
%Y A377466 Cf. A000720, A023055, A031218, A045542, A052410, A053706, A069623, A116086, A116455, A216765, A308658, A336416, A345531, A375740, A376560, A376561, A377057.
%K A377466 nonn,more
%O A377466 1,1
%A A377466 _Gus Wiseman_, Nov 02 2024
%E A377466 a(10) from _Pontus von Brömssen_, Nov 04 2024