This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
We have 125 because 127 is the only prime between 125 and 128.
radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
y=Table[NestWhile[#-1&,Prime[n],radQ[#]&],{n,1000}];
Select[Union[y],Count[y,#]==1&]
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. 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. 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.
N:= 1000: # for terms k where A246655(k+1) <+ N P:= select(isprime,[2,seq(i,i=3..N,2)]): S:= convert(P,set): for p in P while p^2 <= N do S:= S union {seq(p^j,j=2..ilog[p](N))} od: PP:= sort(convert(S,list)): state:= 1: Res:= NULL: ip:= 2: for i from 2 to nops(PP) do if PP[i] = P[ip] then if state = 0 then Res:= Res,i-1 fi; state:= 1; ip:= ip+1; else if state = 1 then Res:= Res,i-1 fi; state:= 0; fi od: Res; # Robert Israel, Jan 22 2025
v=Select[Range[100],PrimePowerQ]; Select[Range[Length[v]-1],Length[Select[Range[v[[#]],v[[#+1]]],PrimeQ]]==1&]
The next prime power after 32 is 37, with interval (32,33,34,35,36,37) containing just one prime 37, so 32 is in the sequence.
v=Select[Range[100],PrimePowerQ] nextpripow[n_]:=NestWhile[#+1&,n+1,!PrimePowerQ[#]&] Select[v,Length[Select[Range[#,nextpripow[#]],PrimeQ]]==1&]
The 15th and 16th perfect powers are 125 and 128, and 127 is the only prime between them, so 15 is in the sequence.
perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1; v=Select[Range[1000],perpowQ]; Select[Range[Length[v]-1],Length[Select[Range[v[[#]],v[[#+1]]],PrimeQ]]==1&]
The prime before 17 is 13, and the interval (13,14,15,16,17) contains only one perfect power 16, so 17 is in the sequence. The prime before 29 is 23, and the interval (23,24,25,26,27,28,29) contains two perfect powers 25 and 27, so 29 is not in the sequence.
perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1; Select[Range[1000],PrimeQ[#]&&Length[Select[Range[NextPrime[#,-1],#],perpowQ]]==1&]
The prime after 13 is 17, and the interval (13,14,15,16,17) contains only one perfect power 16, so 13 is in the sequence.
N:= 10^4: # to get all entries <= N
S:={seq(seq(a^b, b = 2 .. floor(log[a](N))), a = 2 .. floor(sqrt(N)))}:
S:= sort(convert(S,list)):
J:= select(i -> nextprime(S[i]) < S[i+1] and prevprime(S[i]) > S[i-1], [$2..nops(S)-1]):
J:= [1,op(J)]:
map(prevprime, S[J]); # Robert Israel, Jan 19 2025
perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1; Select[Range[1000],PrimeQ[#]&&Length[Select[Range[#,NextPrime[#]],perpowQ]]==1&]
is_a379154(n) = isprime(n) && #select(x->ispower(x), [n+1..nextprime(n+1)-1])==1 \\ Hugo Pfoertner, Dec 19 2024
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