cp's OEIS Frontend

A377288 Numbers k such that there are exactly two prime-powers between prime(k)+1 and prime(k+1)-1.

%I A377288 #13 Nov 09 2024 02:33:29
%S A377288 4,9,30,327,3512
%N A377288 Numbers k such that there are exactly two prime-powers between prime(k)+1 and prime(k+1)-1.
%C A377288 Is this sequence finite? For this conjecture see A053706, A080101, A366833.
%C A377288 Any further terms are > 10^12. - _Lucas A. Brown_, Nov 08 2024
%H A377288 Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/A377288.py">Python program</a>.
%F A377288 prime(a(n)) = A053706(n).
%e A377288 Primes 9 and 10 are 23 and 29, and the interval (24, 25, 26, 27, 28) contains the prime-powers 25 and 27, so 9 is in the sequence.
%t A377288 Select[Range[100], Length[Select[Range[Prime[#]+1,Prime[#+1]-1],PrimePowerQ]]==2&]
%Y A377288 The interval from A008864(n) to A006093(n+1) has A046933 elements.
%Y A377288 For powers of 2 instead of primes see A013597, A014210, A014234, A244508, A304521.
%Y A377288 The corresponding primes are A053706.
%Y A377288 The nearest prime-power before prime(n)-1 is A065514, difference A377289.
%Y A377288 The nearest prime-power after prime(n)+1 is A345531, difference A377281.
%Y A377288 These are the positions of 2 in A080101, or 3 in A366833.
%Y A377288 For at least one prime-power we have A377057, primes A053607.
%Y A377288 For no prime-powers we have A377286.
%Y A377288 For exactly one prime-power we have A377287.
%Y A377288 For squarefree instead of prime-power see A377430, A061398, A377431, A068360.
%Y A377288 A000015 gives the least prime-power >= n.
%Y A377288 A000040 lists the primes, differences A001223.
%Y A377288 A000961 lists the powers of primes, differences A057820.
%Y A377288 A031218 gives the greatest prime-power <= n.
%Y A377288 A246655 lists the prime-powers not including 1, complement A361102.
%Y A377288 Cf. A001597, A002808, A024619, A053707, A064113, A065890, A075526, A095195, A224363, A276781, A376596, A376597, A377282.
%K A377288 nonn,more
%O A377288 1,1
%A A377288 _Gus Wiseman_, Oct 25 2024