%I #23 Sep 12 2024 12:42:20
%S 9,25,27,121,125,243,289,961,1331,1681,2187,3125,3481,4489,4913,6889,
%T 11881,16129,24649,29791,32041,36481,44521,58081,68921,76729,78125,
%U 80089,109561,124609,134689,160801,161051,177147,185761,205379,212521,259081,299209
%N Numbers of the form prime(p)^q where p and q are primes. Prime powers whose prime index and exponent are both prime.
%C Alternatively, numbers of the form prime(prime(i))^prime(j) for some positive integers i, j.
%H Robert Israel, <a href="/A352519/b352519.txt">Table of n, a(n) for n = 1..10000</a>
%e The terms together with their prime indices begin:
%e 9: {2,2}
%e 25: {3,3}
%e 27: {2,2,2}
%e 121: {5,5}
%e 125: {3,3,3}
%e 243: {2,2,2,2,2}
%e 289: {7,7}
%e 961: {11,11}
%e 1331: {5,5,5}
%e 1681: {13,13}
%e 2187: {2,2,2,2,2,2,2}
%e 3125: {3,3,3,3,3}
%e 3481: {17,17}
%e 4489: {19,19}
%e 4913: {7,7,7}
%e 6889: {23,23}
%e 11881: {29,29}
%e 16129: {31,31}
%e 24649: {37,37}
%e 29791: {11,11,11}
%p N:= 10^7: # for terms <= N
%p M:=numtheory:-pi(numtheory:-pi(isqrt(N))):
%p PP:= {seq(ithprime(ithprime(i)),i=1..M)}:
%p R:= NULL:
%p for p in PP do
%p q:= 1:
%p do
%p q:= nextprime(q);
%p t:= p^q;
%p if t > N then break fi;
%p R:= R, t;
%p od;
%p od:
%p sort([R]); # _Robert Israel_, Dec 08 2022
%t Select[Range[10000],PrimePowerQ[#]&&MatchQ[FactorInteger[#],{{_?(PrimeQ[PrimePi[#]]&),k_?PrimeQ}}]&]
%o (Python)
%o from sympy import primepi, integer_nthroot, primerange
%o def A352519(n):
%o def f(x): return int(n+x-sum(primepi(primepi(integer_nthroot(x,p)[0])) for p in primerange(x.bit_length())))
%o def bisection(f,kmin=0,kmax=1):
%o while f(kmax) > kmax: kmax <<= 1
%o while kmax-kmin > 1:
%o kmid = kmax+kmin>>1
%o if f(kmid) <= kmid:
%o kmax = kmid
%o else:
%o kmin = kmid
%o return kmax
%o return bisection(f,n,n) # _Chai Wah Wu_, Sep 12 2024
%Y Numbers of the form p^q for p and q prime are A053810, counted by A001221.
%Y These partitions are counted by A230595.
%Y This is the prime power case of A346068.
%Y For numbers that are not a prime power we have A352518, counted by A352493.
%Y A000040 lists the primes.
%Y A000961 lists prime powers.
%Y A001597 lists perfect powers.
%Y A001694 lists powerful numbers, counted by A007690.
%Y A056166 = prime exponents are all prime, counted by A055923.
%Y A076610 = prime indices are all prime, counted by A000607, powerful A339218.
%Y A109297 = same indices as exponents, counted by A114640.
%Y A112798 lists prime indices, reverse A296150, sum A056239.
%Y A124010 gives prime signature, sorted A118914, sum A001222.
%Y A164336 lists all possible power-towers of prime numbers.
%Y A257994 counts prime indices that are themselves prime, nonprime A330944.
%Y A325131 = disjoint indices from exponents, counted by A114639.
%Y Cf. A000720, A002035, A005117, A007821, A038499, A101436, A181819, A181821.
%K nonn
%O 1,1
%A _Gus Wiseman_, Mar 26 2022