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&]
6 is the largest squarefree less than the 4th prime, 7. So a(4) = 6.
with(numtheory): a:=proc(n) local p,B,j: p:=ithprime(n): B:={}: for j from 1 to p-1 do if abs(mobius(j))>0 then B:=B union {j} else B:=B fi od: B[nops(B)] end: seq(a(m),m=1..75); # Emeric Deutsch, Oct 14 2005
With[{k = 120}, Table[SelectFirst[Range[Prime@ n - 1, Prime@ n - Min[Prime@ n - 1, k], -1], SquareFreeQ], {n, 60}]] (* Michael De Vlieger, Aug 16 2017 *)
a(n,p=prime(n))=while(!issquarefree(p--),); p \\ Charles R Greathouse IV, Aug 16 2017
lpp[n_]:=Module[{k=n-1},While[!PrimePowerQ[k],k--];k]; Join[{1},Table[ lpp[ n],{n,Prime[Range[2,60]]}]] (* Harvey P. Dale, Nov 24 2018 *)
from sympy import factorint, prime def A065514(n): return next(filter(lambda m:len(factorint(m))<=1, range(prime(n)-1,0,-1))) # Chai Wah Wu, Oct 25 2024
The first number-line below shows the perfect powers. The second shows each positive integer k at position prime(k). -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; Select[Range[0,100],#==0||Length[Select[Range[Prime[#]+1,Prime[#+1]-1],perpowQ]]>0&]
The first number line below shows the perfect powers. The second shows each prime. -1-----4-------8-9------------16----------------25--27--------32------36------------------------49-- ===2=3===5===7======11==13======17==19======23==========29==31==========37======41==43======47======
radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Table[NestWhile[#+1&,Prime[n],radQ[#]&],{n,100}]
f(p) = p++; while(!ispower(p), p++); p; lista(nn) = apply(f, primes(nn)); \\ Michel Marcus, Dec 19 2024
The first number line below shows the perfect powers. The second shows each prime. To get a(n) we count the primes between consecutive perfect powers, skipping 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==
N:= 10^6: # to use perfect powers up to N
PP:= {1,seq(seq(i^j,j=2..ilog[i](N)),i=2..isqrt(N))}:
PP:= sort(convert(PP,list)):
M:= map(numtheory:-pi, PP):
subs(0=NULL, M[2..-1]-M[1..-2]): # Robert Israel, Jan 23 2025
radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Length/@Split[Table[NestWhile[#+1&,Prime[n],radQ[#]&],{n,100}]]
The first number line below shows the perfect-powers. The second shows the primes. The third is a(n).
-1-----4-------8-9------------16----------------25--27--------32------36----
===2=3===5===7======11==13======17==19======23==========29==31==========37==
4 8 16 25 32
The terms together with their prime indices begin:
4: {1,1}
8: {1,1,1}
16: {1,1,1,1}
25: {3,3}
32: {1,1,1,1,1}
49: {4,4}
64: {1,1,1,1,1,1}
81: {2,2,2,2}
100: {1,1,3,3}
121: {5,5}
128: {1,1,1,1,1,1,1}
144: {1,1,1,1,2,2}
169: {6,6}
196: {1,1,4,4}
216: {1,1,1,2,2,2}
225: {2,2,3,3}
243: {2,2,2,2,2}
256: {1,1,1,1,1,1,1,1}
radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Union[Table[NestWhile[#+1&,Prime[n],radQ[#]&],{n,100}]]
Table[PrimePi[NextPrime[n]],{n,Select[Range[1000],perpowQ]}]
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 consecutive perfect powers 125 and 128 have interval (125, 126, 127, 128) with unique prime 127, so 128 is in the sequence.
radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
y=Table[NestWhile[#+1&,Prime[n],radQ[#]&],{n,1000}];
Select[Union[y],Count[y,#]==1&]
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