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A060845 Largest prime < a nontrivial power of a prime.

%I A060845 #18 Sep 15 2024 22:00:16
%S A060845 3,7,7,13,23,23,31,47,61,79,113,113,127,167,241,251,283,337,359,509,
%T A060845 523,619,727,839,953,1021,1327,1367,1669,1847,2039,2179,2179,2207,
%U A060845 2399,2803,3121,3469,3719,4093,4483,4909,5039,5323,6229,6553,6857,6883,7919
%N A060845 Largest prime < a nontrivial power of a prime.
%H A060845 Harry J. Smith, <a href="/A060845/b060845.txt">Table of n, a(n) for n = 1..1000</a>
%F A060845 a(n) = prevprime[A025475(n)] = A007917[A025475(n)] = Max{p| p < A025475(n)}
%e A060845 78125=5^7 follows 78121
%t A060845 Take[NextPrime[#,-1]&/@Union[Flatten[Table[Prime[p]^n,{n,2,20},{p,25}]]], 50] (* _Harvey P. Dale_, Mar 26 2012 *)
%o A060845 (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
%o A060845 (Python)
%o A060845 from sympy import primepi, integer_nthroot, prevprime
%o A060845 def A060845(n):
%o A060845     def bisection(f,kmin=0,kmax=1):
%o A060845         while f(kmax) > kmax: kmax <<= 1
%o A060845         while kmax-kmin > 1:
%o A060845             kmid = kmax+kmin>>1
%o A060845             if f(kmid) <= kmid:
%o A060845                 kmax = kmid
%o A060845             else:
%o A060845                 kmin = kmid
%o A060845         return kmax
%o A060845     def f(x): return int(n+x-sum(primepi(integer_nthroot(x,k)[0]) for k in range(2,x.bit_length())))
%o A060845     return prevprime(bisection(f,n,n)) # _Chai Wah Wu_, Sep 15 2024
%Y A060845 Cf. A025475, A000961, A001597, A001694, A007917, A007918, A013632, A013633, A049711, A060846.
%K A060845 nonn
%O A060845 1,1
%A A060845 _Labos Elemer_, May 03 2001