A377288 - OEIS

%I #13 Nov 09 2024 02:33:29

%S 4,9,30,327,3512

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

%C Is this sequence finite? For this conjecture see A053706, A080101, A366833.

%C Any further terms are > 10^12. - _Lucas A. Brown_, Nov 08 2024

%H Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/A377288.py">Python program</a>.

%F prime(a(n)) = A053706(n).

%e 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 Select[Range[100], Length[Select[Range[Prime[#]+1,Prime[#+1]-1],PrimePowerQ]]==2&]

%Y The interval from A008864(n) to A006093(n+1) has A046933 elements.

%Y For powers of 2 instead of primes see A013597, A014210, A014234, A244508, A304521.

%Y The corresponding primes are A053706.

%Y The nearest prime-power before prime(n)-1 is A065514, difference A377289.

%Y The nearest prime-power after prime(n)+1 is A345531, difference A377281.

%Y These are the positions of 2 in A080101, or 3 in A366833.

%Y For at least one prime-power we have A377057, primes A053607.

%Y For no prime-powers we have A377286.

%Y For exactly one prime-power we have A377287.

%Y For squarefree instead of prime-power see A377430, A061398, A377431, A068360.

%Y A000015 gives the least prime-power >= n.

%Y A000040 lists the primes, differences A001223.

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

%Y A031218 gives the greatest prime-power <= n.

%Y A246655 lists the prime-powers not including 1, complement A361102.

%Y Cf. A001597, A002808, A024619, A053707, A064113, A065890, A075526, A095195, A224363, A276781, A376596, A376597, A377282.

%K nonn,more

%O 1,1

%A _Gus Wiseman_, Oct 25 2024