A352519 Numbers of the form prime(p)^q where p and q are primes. Prime powers whose prime index and exponent are both prime.

9, 25, 27, 121, 125, 243, 289, 961, 1331, 1681, 2187, 3125, 3481, 4489, 4913, 6889, 11881, 16129, 24649, 29791, 32041, 36481, 44521, 58081, 68921, 76729, 78125, 80089, 109561, 124609, 134689, 160801, 161051, 177147, 185761, 205379, 212521, 259081, 299209

Offset: 1

Author

Gus Wiseman, Mar 26 2022

Keywords

• nonn

Comments

Alternatively, numbers of the form prime(prime(i))^prime(j) for some positive integers i, j.

Examples

			The terms together with their prime indices begin:
      9: {2,2}
     25: {3,3}
     27: {2,2,2}
    121: {5,5}
    125: {3,3,3}
    243: {2,2,2,2,2}
    289: {7,7}
    961: {11,11}
   1331: {5,5,5}
   1681: {13,13}
   2187: {2,2,2,2,2,2,2}
   3125: {3,3,3,3,3}
   3481: {17,17}
   4489: {19,19}
   4913: {7,7,7}
   6889: {23,23}
  11881: {29,29}
  16129: {31,31}
  24649: {37,37}
  29791: {11,11,11}
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Numbers of the form p^q for p and q prime are A053810, counted by A001221.

These partitions are counted by A230595.

This is the prime power case of A346068.

For numbers that are not a prime power we have A352518, counted by A352493.

A000040 lists the primes.

A000961 lists prime powers.

A001597 lists perfect powers.

A001694 lists powerful numbers, counted by A007690.

A056166 = prime exponents are all prime, counted by A055923.

A076610 = prime indices are all prime, counted by A000607, powerful A339218.

A109297 = same indices as exponents, counted by A114640.

A112798 lists prime indices, reverse A296150, sum A056239.

A124010 gives prime signature, sorted A118914, sum A001222.

A164336 lists all possible power-towers of prime numbers.

A257994 counts prime indices that are themselves prime, nonprime A330944.

A325131 = disjoint indices from exponents, counted by A114639.

Cf. A000720, A002035, A005117, A007821, A038499, A101436, A181819, A181821.

Programs

Maple

N:= 10^7: # for terms <= N
M:=numtheory:-pi(numtheory:-pi(isqrt(N))):
PP:= {seq(ithprime(ithprime(i)),i=1..M)}:
R:= NULL:
for p in PP do
  q:= 1:
  do
    q:= nextprime(q);
    t:= p^q;
    if t > N then break fi;
    R:= R, t;
  od;
od:
sort([R]); # Robert Israel, Dec 08 2022

Mathematica

Select[Range[10000],PrimePowerQ[#]&&MatchQ[FactorInteger[#],{{?(PrimeQ[PrimePi[#]]&),k?PrimeQ}}]&]

Python

from sympy import primepi, integer_nthroot, primerange
def A352519(n):
    def f(x): return int(n+x-sum(primepi(primepi(integer_nthroot(x,p)[0])) for p in primerange(x.bit_length())))
    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
    return bisection(f,n,n) # Chai Wah Wu, Sep 12 2024