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A377287 Numbers k such that there is exactly one prime-power between prime(k)+1 and prime(k+1)-1.

%I A377287 #11 Oct 28 2024 12:05:24
%S A377287 2,6,11,15,18,22,31,39,53,54,61,68,72,97,99,114,129,146,162,172,217,
%T A377287 219,263,283,309,329,357,409,445,487,519,564,609,656,675,705,811,847,
%U A377287 882,886,1000,1028,1163,1252,1294,1381,1423,1457,1523,1715,1821,1877,1900
%N A377287 Numbers k such that there is exactly one prime-power between prime(k)+1 and prime(k+1)-1.
%e A377287 Primes 18 and 19 are 61 and 67, and the interval (62, 63, 64, 65, 66) contains only the one prime-power 64, so 18 is in the sequence.
%t A377287 Select[Range[100],Length[Select[Range[Prime[#]+1,Prime[#+1]-1],PrimePowerQ]]==1&]
%o A377287 (Python)
%o A377287 from itertools import count, islice
%o A377287 from sympy import factorint, nextprime
%o A377287 def A377287_gen(): # generator of terms
%o A377287     p, q, k = 2, 3, 1
%o A377287     for k in count(1):
%o A377287         if sum(1 for i in range(p+1,q) if len(factorint(i))<=1)==1:
%o A377287             yield k
%o A377287         p, q = q, nextprime(q)
%o A377287 A377287_list = list(islice(A377287_gen(),53)) # _Chai Wah Wu_, Oct 28 2024
%Y A377287 The interval from A008864(n) to A006093(n+1) has A046933(n) elements.
%Y A377287 For powers of 2 instead of primes see A013597, A014210, A014234, A244508, A304521.
%Y A377287 The nearest prime-power before prime(n)-1 is A065514, difference A377289.
%Y A377287 The nearest prime-power after prime(n)+1 is A345531, difference A377281.
%Y A377287 These are the positions of 1 in A080101, or 2 in A366833.
%Y A377287 For at least one prime-power we have A377057, primes A053607.
%Y A377287 For no prime-powers we have A377286.
%Y A377287 For two prime-powers we have A377288, primes A053706.
%Y A377287 For squarefree instead of prime-power see A377430, A061398, A377431, A068360.
%Y A377287 A000015 gives the least prime-power >= n.
%Y A377287 A000040 lists the primes, differences A001223.
%Y A377287 A000961 lists the powers of primes, differences A057820.
%Y A377287 A031218 gives the greatest prime-power <= n.
%Y A377287 A246655 lists the prime-powers not including 1, complement A361102.
%Y A377287 Cf. A001597, A002808, A024619, A053707, A064113, A065890, A075526, A095195, A224363, A276781, A376597, A377282.
%K A377287 nonn
%O A377287 1,1
%A A377287 _Gus Wiseman_, Oct 25 2024