A116455 -id:A116455 - cp's OEIS frontend

A116086 Perfect powers n with no primes between n and the next larger perfect power, which is in A116455.

8, 25, 32, 121, 2187, 3125, 32761, 79507, 97336, 503284356

Offset: 1

T. D. Noe, Mar 28 2006

No other n<10^12. There is a conjecture that this sequence is finite.
No other terms < 10^18. - Jud McCranie, Nov 03 2013
No other terms < 4.5*10^18. - Giovanni Resta, Apr 28 2014

			The prime-free ranges are (2^3,3^2), (5^2,3^3), (2^5,6^2), (11^2,5^3), (3^7,13^3), (5^5,56^2), (181^2,2^15), (43^3,282^2), (46^3,312^2), (22434^2,55^5).

March 2006 NMBRTHRY Archives
PlanetMath, Redmond-Sun conjecture

Cf. A001597 (perfect powers), A116455.

Cf. A068435 (for prime powers).

lim=10^12; lst={}; k=2; While[n=Floor[lim^(1/k)]; n>=2, lst=Join[lst,Range[2,n]^k]; k++ ]; lst=Union[lst]; PrimeFree[n1_,n2_] := Module[{n=n1+1}, While[n

A377466 Numbers k such that there is more than one perfect power x in the range prime(k) < x < prime(k+1).

Original entry on oeis.org

4, 9, 11, 30, 327, 445, 3512, 7789, 9361, 26519413

Offset: 1

Gus Wiseman, Nov 02 2024

Perfect powers (A001597) are numbers with a proper integer root, the complement of A007916.
Is this sequence finite?
The Redmond-Sun conjecture (see A308658) implies that this sequence is finite. - Pontus von Brömssen, Nov 05 2024

			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.

For powers of 2 see A013597, A014210, A014234, A188951, A244508, A377467.
For no prime-powers we have A377286, ones in A080101.
For a unique prime-power we have A377287.
For squarefree numbers see A377430, A061398, A377431, A068360, A224363.
These are the positions of terms > 1 in A377432.
For a unique perfect power we have A377434.
For no perfect powers we have A377436.
A000015 gives the least prime power >= n.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A001597 lists the perfect powers, differences A053289, seconds A376559.
A007916 lists the non-perfect-powers, differences A375706, seconds A376562.
A046933 counts the interval from A008864(n) to A006093(n+1).
A081676 gives the greatest perfect power <= n.
A131605 lists perfect powers that are not prime-powers.
A246655 lists the prime-powers not including 1, complement A361102.
A366833 counts prime-powers between primes, see A053607, A304521.
A377468 gives the least perfect power > n.
Cf. A000720, A023055, A031218, A045542, A052410, A053706, A069623, A116086, A116455, A216765, A308658, A336416, A345531, A375740, A376560, A376561, A377057.

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

a(n) = A000720(A116086(n)) = A000720(A116455(n)) for n <= 10. This would hold for all n if there do not exist more than two perfect powers between any two consecutive primes, which is implied by the Redmond-Sun conjecture. - Pontus von Brömssen, Nov 05 2024

a(10) from Pontus von Brömssen, Nov 04 2024

A068435 Consecutive prime powers without a prime between them.

Original entry on oeis.org

8, 9, 25, 27, 121, 125, 2187, 2197, 32761, 32768

Offset: 1

Jon Perry, Mar 09 2002

From David A. Corneth, Aug 24 2019: (Start)
Only 5 pairs are known up to 4*10^18. Legendre's conjecture states that there is a prime number between n^2 and (n + 1)^2 for every positive integer n. The conjecture has been verified up to n = 2*10^9. So to that bound we only have to check for two prime powers where at least one has an exponent of at least 3. That has been done to prime powers <= 10^22.
If there is another pair besides the first five listed with both numbers <= 10^22 then Legendre's conjecture is false.
Proof: If there is another such pair with both numbers <= 10^22 then it must be of the form [p^2, q^2] where p is a prime and q is the least prime larger than p. Then q - p >= 2 (as p != 2). So there is no prime between p^2 and q^2 and hence there is no prime between p^2 and (p+1)^2. This is a counterexample to Legendre's conjecture. (End)

			8 = 2^3, 9 = 3^2, there is no prime between 8 and 9.
25 = 5^2, 27 = 3^3, there is no prime between 25 and 27.

David A. Corneth, PARI program to search for pairs where not both numbers are squares of primes
Wikipedia, Legendre's conjecture

Cf. A014085, A025475, A060846, A067871, A068315, A246547, A366835.

Cf. A116086 and A116455 (for perfect powers, but not necessarily prime powers).

With[{upto=33000},Select[Partition[Select[Range[upto],PrimePowerQ],2,1],NoneTrue[#,PrimeQ]&]] (* Paolo Xausa, Oct 29 2023 *)

ispp(x) = !isprime(x) && isprimepower(x);
lista(nn=50000) = {my(prec = 0); for (i=1, nn, if (ispp(i), if (! prec, prec = i, if (primepi(i) == primepi(prec), print1(prec, ", ", i, ", ")); prec = i;);););} \\ Michel Marcus, Aug 24 2019

A293190 a(n) = |{A001597(n) <= k <= A001597(n+1): 2*k^2-1 is prime}|.

Original entry on oeis.org

3, 4, 1, 4, 6, 1, 1, 2, 9, 8, 6, 7, 7, 1, 3, 6, 8, 11, 8, 1, 6, 5, 11, 14, 4, 2, 12, 14, 16, 8, 6, 15, 13, 9, 16, 16, 15, 15, 13, 10, 6, 16, 21, 16, 11, 4, 8, 22, 23, 17, 20, 7, 8, 23, 18, 21, 4, 23, 13, 1, 4, 24, 28, 24, 24, 24, 8, 14, 23, 24, 25, 1, 24, 15, 2, 21, 29, 26, 24, 35, 27, 25, 31, 30, 31, 30, 24, 4, 30, 30, 32, 30, 35, 31, 13, 13, 33, 31, 29, 31

Offset: 1

Zhi-Wei Sun, Oct 01 2017

Conjecture: (i) a(n) > 0 for all n > 0. In other words, for any perfect powers x^m and y^n with 0 < x^m < y^n, there is an integer z with x^m <= z <= y^n such that 2*z^2 - 1 is prime.
(ii) For any perfect powers x^m and y^n with 0 < x^m < y^n, there is an integer z with x^m <= z <= y^n such that 2*z + 3 (or 20*z^2 + 3) is prime.
(iii) For perfect powers x^m and y^n with 0 < x^m < y^n, there is a practical number q (cf. A005153) with x^m <= q <= y^n, unless x^m = 5^2 and y^n = 3^3, or x^m = 11^2 and y^n = 5^3, or x^m = 22434^2 and y^n = 55^5.
Compare this with the Redmond-Sun conjecture.

			a(1) = 3 since 2*2^2 - 1, 2*3^2-1 and 2*4^2-1 are all prime but 2*1^2 - 1 is not prime.
a(3) = 1 since A001597(3) = 8, A001597(4) = 9, 2*8^2 - 1 = 127 is prime but 2*9^2 - 1 is composite.
a(6) = 1 since A001597(6) = 25, A001597(7) = 27, 2*25^2 - 1 = 1249 is prime but 2*26^2 - 1 and 2*27^2 - 1 are composite.
a(14) = 1 since A001597(14) = 121, A001597(15) = 125, 2*125^2
- 1 = 31249 is prime but 2*k^2 - 1 is composite for every k = 121, 122, 123, 124.
a(361) = 1 since A001597(361) = 46^3 = 97336, A001597(362) = 312^2 = 97344, and k = 97342 is the only number among 97336,...,97344 with 2*k^2 - 1 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..5000
Wikipedia, Redmond-Sun conjecture

Cf. A001597, A005153, A066049, A116086, A116455.

n=1;m=1;Do[Do[If[IntegerQ[k^(1/Prime[i])],Print[n," ",Sum[Boole[PrimeQ[2j^2-1]],{j,m,k}]];n=n+1;m=k;Goto[aa]],{i,1,PrimePi[Log[2,k]]}];Label[aa],{k,2,6561}]

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A116086 Perfect powers n with no primes between n and the next larger perfect power, which is in A116455.

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A377466 Numbers k such that there is more than one perfect power x in the range prime(k) < x < prime(k+1).

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A068435 Consecutive prime powers without a prime between them.

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A293190 a(n) = |{A001597(n) <= k <= A001597(n+1): 2*k^2-1 is prime}|.

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