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 first number line below shows the perfect powers. The second shows each prime. To get a(n), we take the last perfect power in each interval between consecutive primes, omitting the cases where there are none. -1-----4-------8-9------------16----------------25--27--------32------36---- ===2=3===5===7======11==13======17==19======23==========29==31==========37==
radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Union[Table[NestWhile[#-1&,Prime[n],radQ[#]&],{n,1000}]]
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 first number line below shows the perfect powers. The second shows each n at position prime(n). To get a(n), we take the first prime between each pair of consecutive perfect powers, skipping the cases where there are none. -1-----4-------8-9------------16----------------25--27--------32------36---- ===1=2===3===4=======5===6=======7===8=======9==========10==11==========12==
perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Union[1+Table[PrimePi[n],{n,Select[Range[100],perpowQ]}]]
Between prime(4) = 7 and prime(5) = 11 the only non-perfect-power is 10, so a(4) = 1.
radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Table[Length[Select[Range[Prime[n]+1, Prime[n+1]-1],radQ]],{n,100}]
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
The composite numbers counted by a(n) cover A106543 with the following disjoint sets: . 6 . 10 12 14 15 18 20 21 22 24 26 28 30 33 34 35 38 39 40 42 44 45 46 48 50 51 52 54 55 56 57 58 60 62 63
perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
v=Select[Range[100],perpowQ[#]&];
Table[Length[Select[Range[v[[i]]+1,v[[i+1]]-1],CompositeQ]],{i,Length[v]-1}]
from sympy import mobius, integer_nthroot, primepi def A378614(n): def bisection(f,kmin=0,kmax=1): while f(kmax) > kmax: kmax <<= 1 while kmax-kmin > 1: kmid = kmax+kmin>>1 if f(kmid) <= kmid: kmax = kmid else: kmin = kmid return kmax def f(x): return int(n+x-1+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length()))) return -(a:=bisection(f,n,n))+(b:=bisection(lambda x:f(x)+1,a+1,a+1))-primepi(b)+primepi(a)-1 # Chai Wah Wu, Dec 03 2024
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