cp's OEIS Frontend

A377057 Numbers k such that there is at least one prime-power between prime(k)+1 and prime(k+1)-1.

%I A377057 #11 Oct 28 2024 04:51:26
%S A377057 2,4,6,9,11,15,18,22,30,31,39,53,54,61,68,72,97,99,114,129,146,162,
%T A377057 172,217,219,263,283,309,327,329,357,409,445,487,519,564,609,656,675,
%U A377057 705,811,847,882,886,1000,1028,1163,1252,1294,1381,1423,1457
%N A377057 Numbers k such that there is at least one prime-power between prime(k)+1 and prime(k+1)-1.
%F A377057 prime(a(n)) = A053607(n).
%e A377057 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 A377057 Select[Range[100], Length[Select[Range[Prime[#]+1,Prime[#+1]-1],PrimePowerQ]]>=1&]
%o A377057 (Python)
%o A377057 from itertools import count, islice
%o A377057 from sympy import factorint, nextprime
%o A377057 def A377057_gen(): # generator of terms
%o A377057     p, q, k = 2, 3, 1
%o A377057     for k in count(1):
%o A377057         if any(len(factorint(i))<=1 for i in range(p+1,q)):
%o A377057             yield k
%o A377057         p, q = q, nextprime(q)
%o A377057 A377057_list = list(islice(A377057_gen(),52)) # _Chai Wah Wu_, Oct 27 2024
%Y A377057 The interval from A008864(n) to A006093(n+1) has A046933(n) elements.
%Y A377057 For powers of 2 instead of primes see A013597, A014210, A014234, A244508, A304521.
%Y A377057 The corresponding primes are A053607.
%Y A377057 The nearest prime-power before prime(n)-1 is A065514, difference A377289.
%Y A377057 These are the positions of positive terms in A080101, or terms >1 in A366833.
%Y A377057 The nearest prime-power after prime(n)+1 is A345531, difference A377281.
%Y A377057 For no prime-powers we have A377286.
%Y A377057 For exactly one prime-power we have A377287.
%Y A377057 For exactly two prime-powers we have A377288, primes A053706.
%Y A377057 A000015 gives the least prime-power >= n.
%Y A377057 A000040 lists the primes, differences A001223.
%Y A377057 A000961 lists the powers of primes, differences A057820.
%Y A377057 A031218 gives the greatest prime-power <= n.
%Y A377057 A246655 lists the prime-powers not including 1, complement A361102.
%Y A377057 Cf. A001597, A002808, A024619, A053707, A064113, A065890, A075526, A095195, A224363, A276781, A376597, A377051, A377282.
%K A377057 nonn
%O A377057 1,1
%A A377057 _Gus Wiseman_, Oct 25 2024