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.
Differences[Table[NestWhile[#+1&,Prime[n],SquareFreeQ[#]&],{n,100}]]
Differences[Table[NestWhile[#+1&,n,#>1&&SquareFreeQ[#]&],{n,100}]]
Union[Table[NestWhile[#+1&,Prime[n],SquareFreeQ],{n,100}]]
lns[p_]:=Module[{k=p+1},While[SquareFreeQ[k],k++];k]; Table[lns[p],{p,Prime[Range[70]]}]//Union (* Harvey P. Dale, Jun 12 2025 *)
Primes 9 and 10 are 23 and 29, and the interval (24,25,26,27,28) contains two perfect powers (25,27), so 9 is in the sequence.
perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1; Select[Range[100],Count[Range[Prime[#]+1, Prime[#+1]-1],_?perpowQ]>1&]
from itertools import islice from sympy import prime from gmpy2 import is_power, next_prime def A377466_gen(startvalue=1): # generator of terms >= startvalue k = max(startvalue,1) p = prime(k) while (q:=next_prime(p)): c = 0 for i in range(p+1,q): if is_power(i): c += 1 if c>1: yield k break k += 1 p = q A377466_list = list(islice(A377466_gen(),9)) # Chai Wah Wu, Nov 04 2024
The first few prime powers A246547 are 4, 8, 9, 16. The first few primes are 2, 3, 5, 7, 11, 13. We have (4), 5, 7, (8), (9), 11, 13, (16) and so the sequence begins with 2, 0, 2.
The initial terms count the following sets of primes: {5,7}, {}, {11,13}, {17,19,23}, {}, {29,31}, {37,41,43,47}, ... - _Gus Wiseman_, Dec 02 2024
t = {}; cnt = 0; Do[If[PrimePowerQ[n], If[FactorInteger[n][[1, 2]] == 1, cnt++, AppendTo[t, cnt]; cnt = 0]], {n, 4 + 1, 30000}]; t (* T. D. Noe, May 21 2013 *)
nn = 2^20; Differences@ Map[PrimePi, Select[Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], PrimePowerQ] ] (* Michael De Vlieger, Oct 26 2023 *)
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&]
Primes 9 and 10 are 23 and 29, and the interval (24, 25, 26, 27, 28) contains the prime-powers 25 and 27, so 9 is in the sequence.
Select[Range[100], Length[Select[Range[Prime[#]+1,Prime[#+1]-1],PrimePowerQ]]==2&]
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==
radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Table[NestWhile[#-1&,Prime[n],radQ[#]&],{n,100}]
a(n) = my(k=prime(n)-1); while (!(ispower(k) || (k==1)), k--); k; \\ Michel Marcus, Nov 25 2024
from sympy import mobius, integer_nthroot, prime def A378035(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(x-1+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length()))) m = (p:=prime(n)-1)-f(p) return bisection(lambda x:f(x)+m,m,m) # Chai Wah Wu, Nov 25 2024
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 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}]]
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