A060845 Largest prime < a nontrivial power of a prime.

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

Author

Labos Elemer, May 03 2001

Keywords

• nonn

Examples

			78125=5^7 follows 78121
		

Links

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

Crossrefs

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

Programs

Mathematica

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

PARI

{ 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

Python

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

Formula

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