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

1,1

Comments

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

Links

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

Example

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}

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}}]&]

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.

Sequence in context: A074852 A322177 A346068 * A321874 A020252 A076486

Adjacent sequences: A352516 A352517 A352518 * A352520 A352521 A352522

Keyword

nonn

Author

Gus Wiseman, Mar 26 2022

Status

approved