cp's OEIS Frontend

A377286 Numbers k such that there are no prime-powers between prime(k)+1 and prime(k+1)-1.

%I A377286 #10 Oct 27 2024 13:03:43
%S A377286 1,3,5,7,8,10,12,13,14,16,17,19,20,21,23,24,25,26,27,28,29,32,33,34,
%T A377286 35,36,37,38,40,41,42,43,44,45,46,47,48,49,50,51,52,55,56,57,58,59,60,
%U A377286 62,63,64,65,66,67,69,70,71,73,74,75,76,77,78,79,80,81,82
%N A377286 Numbers k such that there are no prime-powers between prime(k)+1 and prime(k+1)-1.
%e A377286 Primes 18 and 19 are 61 and 67, and the interval (62, 63, 64, 65, 66) contains the prime-power 64, so 18 is not in the sequence.
%t A377286 Select[Range[100], Length[Select[Range[Prime[#]+1,Prime[#+1]-1],PrimePowerQ]]==0&]
%o A377286 (Python)
%o A377286 from itertools import count, islice
%o A377286 from sympy import factorint, nextprime
%o A377286 def A377286_gen(): # generator of terms
%o A377286     p, q, k = 2, 3, 1
%o A377286     for k in count(1):
%o A377286         if all(len(factorint(i))>1 for i in range(p+1,q)):
%o A377286             yield k
%o A377286         p, q = q, nextprime(q)
%o A377286 A377286_list = list(islice(A377286_gen(),66)) # _Chai Wah Wu_, Oct 27 2024
%Y A377286 The interval from A008864(n) to A006093(n+1) has A046933(n) elements.
%Y A377286 For powers of 2 instead of primes see A013597, A014210, A014234, A244508, A304521.
%Y A377286 The nearest prime-power before prime(n)-1 is A065514, difference A377289.
%Y A377286 These are the positions of 0 in A080101, or 1 in A366833.
%Y A377286 The nearest prime-power after prime(n)+1 is A345531, difference A377281.
%Y A377286 For at least one prime-power we have A377057.
%Y A377286 For one instead of no prime-powers we have A377287.
%Y A377286 For two instead of no prime-powers we have A377288.
%Y A377286 A000015 gives the least prime-power >= n.
%Y A377286 A000040 lists the primes, differences A001223.
%Y A377286 A000961 lists the powers of primes, differences A057820.
%Y A377286 A031218 gives the greatest prime-power <= n.
%Y A377286 A246655 lists the prime-powers not including 1, complement A361102.
%Y A377286 Cf. A001597, A002808, A024619, A053707, A064113, A065890, A075526, A095195, A276781, A376597, A377051, A377282.
%K A377286 nonn
%O A377286 1,2
%A A377286 _Gus Wiseman_, Oct 25 2024