A060846 -id:A060846 - cp's OEIS frontend

A068435 Consecutive prime powers without a prime between them.

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

A060845 Largest prime < a nontrivial power of a prime.

Original entry on oeis.org

3, 7, 7, 13, 23, 23, 31, 47, 61, 79, 113, 113, 127, 167, 241, 251, 283, 337, 359, 509, 523, 619, 727, 839, 953, 1021, 1327, 1367, 1669, 1847, 2039, 2179, 2179, 2207, 2399, 2803, 3121, 3469, 3719, 4093, 4483, 4909, 5039, 5323, 6229, 6553, 6857, 6883, 7919

Offset: 1

Labos Elemer, May 03 2001

nonn

			78125=5^7 follows 78121

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A025475, A000961, A001597, A001694, A007917, A007918, A013632, A013633, A049711, A060846.

Take[NextPrime[#,-1]&/@Union[Flatten[Table[Prime[p]^n,{n,2,20},{p,25}]]], 50] (* Harvey P. Dale, Mar 26 2012 *)

{ m=1; for (n=1, 1000, m++; while(sigma(m)*eulerphi(m)*(1 - isprime(m)) <= (m - 1)^2, m++); write("b060845.txt", n, " ", precprime(m - 1)); ) } \\ Harry J. Smith, Jul 19 2009

from sympy import primepi, integer_nthroot, prevprime
def A060845(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-sum(primepi(integer_nthroot(x,k)[0]) for k in range(2,x.bit_length())))
    return prevprime(bisection(f,n,n)) # Chai Wah Wu, Sep 15 2024

a(n) = prevprime[A025475(n)] = A007917[A025475(n)] = Max{p| p < A025475(n)}

cp's OEIS Frontend

A068435 Consecutive prime powers without a prime between them.

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