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A379155 Numbers k such that there is a unique prime between the k-th and (k+1)-th prime powers (A246655).

%I A379155 #14 Jan 23 2025 00:22:28
%S A379155 2,3,5,7,9,10,13,15,17,18,22,23,26,27,31,32,40,42,43,44,52,53,67,68,
%T A379155 69,70,77,78,85,86,90,91,116,117,119,120,135,136,151,152,169,170,186,
%U A379155 187,197,198,243,244,246,247,291,292,312,313,339,340,358,360,362
%N A379155 Numbers k such that there is a unique prime between the k-th and (k+1)-th prime powers (A246655).
%C A379155 Numbers k such that exactly one of A246655(k) and A246655(k+1) is prime. - _Robert Israel_, Jan 22 2025
%C A379155 The prime powers themselves are: 3, 4, 7, 9, 13, 16, 23, 27, 31, 32, 47, 49, 61, 64, ...
%H A379155 Robert Israel, <a href="/A379155/b379155.txt">Table of n, a(n) for n = 1..2112</a>
%F A379155 A246655(a(n)) = A379157(n).
%e A379155 The 4th and 5th prime powers are 5 and 7, with interval (5,6,7) containing two primes, so 4 is not in the sequence.
%e A379155 The 13th and 14th prime powers are 23 and 25, with interval (23,24,25) containing only one prime, so 13 is in the sequence.
%e A379155 The 18th and 19th prime powers are 32 and 37, with interval (32,33,34,35,36,37) containing just one prime 37, so 18 is in the sequence.
%p A379155 N:= 1000: # for terms k where A246655(k+1) <+ N
%p A379155 P:= select(isprime,[2,seq(i,i=3..N,2)]):
%p A379155 S:= convert(P,set):
%p A379155 for p in P while p^2 <= N do
%p A379155   S:= S union {seq(p^j,j=2..ilog[p](N))}
%p A379155 od:
%p A379155 PP:= sort(convert(S,list)):
%p A379155 state:= 1: Res:= NULL:
%p A379155 ip:= 2:
%p A379155 for i from 2 to nops(PP) do
%p A379155   if PP[i] = P[ip] then
%p A379155     if state = 0 then Res:= Res,i-1 fi;
%p A379155     state:= 1;
%p A379155     ip:= ip+1;
%p A379155   else
%p A379155     if state = 1 then Res:= Res,i-1 fi;
%p A379155     state:= 0;
%p A379155   fi
%p A379155 od:
%p A379155 Res; # _Robert Israel_, Jan 22 2025
%t A379155 v=Select[Range[100],PrimePowerQ];
%t A379155 Select[Range[Length[v]-1],Length[Select[Range[v[[#]],v[[#+1]]],PrimeQ]]==1&]
%Y A379155 These are the positions of 1 in A366835, for perfect powers A080769.
%Y A379155 For perfect powers instead of prime powers we have A378368.
%Y A379155 For no primes we have A379156, for perfect powers A274605.
%Y A379155 The prime powers themselves are A379157, for previous A175106.
%Y A379155 A000015 gives the least prime power >= n.
%Y A379155 A000040 lists the primes, differences A001223.
%Y A379155 A000961 lists the powers of primes, differences A057820.
%Y A379155 A031218 gives the greatest prime power <= n.
%Y A379155 A065514 gives the greatest prime power < prime(n), difference A377289.
%Y A379155 A246655 lists the prime powers.
%Y A379155 A366833 counts prime powers between primes, see A053607, A304521.
%Y A379155 Cf. A025474, A067871, A068315, A080101, A178700, A345531, A377281, A377283, A377287, A377434, A378374.
%K A379155 nonn
%O A379155 1,1
%A A379155 _Gus Wiseman_, Dec 22 2024