A377057 - OEIS

%I #11 Oct 28 2024 04:51:26

%S 2,4,6,9,11,15,18,22,30,31,39,53,54,61,68,72,97,99,114,129,146,162,

%T 172,217,219,263,283,309,327,329,357,409,445,487,519,564,609,656,675,

%U 705,811,847,882,886,1000,1028,1163,1252,1294,1381,1423,1457

%N Numbers k such that there is at least one prime-power between prime(k)+1 and prime(k+1)-1.

%F prime(a(n)) = A053607(n).

%e Primes 18 and 19 are 61 and 67, and the interval (62, 63, 64, 65, 66) contains the prime-power 64, so 18 is in the sequence.

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

%o (Python)

%o from itertools import count, islice

%o from sympy import factorint, nextprime

%o def A377057_gen(): # generator of terms

%o     p, q, k = 2, 3, 1

%o     for k in count(1):

%o         if any(len(factorint(i))<=1 for i in range(p+1,q)):

%o             yield k

%o         p, q = q, nextprime(q)

%o A377057_list = list(islice(A377057_gen(),52)) # _Chai Wah Wu_, Oct 27 2024

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

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

%Y The corresponding primes are A053607.

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

%Y These are the positions of positive terms in A080101, or terms >1 in A366833.

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

%Y For no prime-powers we have A377286.

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

%Y For exactly two prime-powers we have A377288, primes A053706.

%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, A376597, A377051, A377282.

%K nonn

%O 1,1

%A _Gus Wiseman_, Oct 25 2024