A375733 Integers of the form p^(q^r) where p, q, r are distinct primes.

390625, 1953125, 5764801, 40353607, 214358881, 815730721, 2357947691, 6975757441, 10604499373, 16983563041, 78310985281, 118587876497, 322687697779, 500246412961, 847288609443, 852891037441, 1801152661463, 3512479453921, 7984925229121, 11688200277601, 14507145975869

Offset

1,1

Comments

a(15) is the first term not an eighth or ninth power.

Links

Table of n, a(n) for n=1..21.

Example

390625 is a term because it equals 5^(2^3).
118587876497 is a term because it equals 17^(3^2).

Maple

N:= 10^20: # for terms <= N
with(priqueue):
initialize(pq):
if 2^(3^5) < N then insert([-2^(3^5), 2, 3, 5], pq) fi:
if 2^(5^3) < N then insert([-2^(5^3), 2, 5, 3], pq) fi:
if 3^(2^5) < N then insert([-3^(2^5), 3, 2, 5], pq) fi:
if 3^(5^2) < N then insert([-3^(5^2), 3, 5, 2], pq) fi:
if 5^(2^3) < N then insert([-5^(2^3), 5, 2, 3], pq) fi:
if 5^(3^2) < N then insert([-5^(3^2), 5, 3, 2], pq) fi:
Res:= NULL:
while pq[0] > 0 do
  t:= extract(pq);
  Res:= Res, -t[1];
  q:= nextprime(t[2]);
  while member(q, {t[3], t[4]}) do q:= nextprime(q) od;
  v:= q^(t[3]^t[4]);
  if v <= N then insert([-v, q, t[3], t[4]], pq) fi;
  if t[2] = 2 then
    q:= nextprime(t[3]);
    if q = t[4] then q:= nextprime(q) fi;
    v:= 2^(q^t[4]);
    if v <= N then insert([-v, 2, q, t[4]], pq) fi;
  fi;
  if {t[2], t[3]} = {2, 3} then
    q:= nextprime(t[4]);
    v:= t[2]^(t[3]^q);
    if v <= N then insert([-v, t[2], t[3], q], pq) fi;
  fi
od:
Res; # Robert Israel, Aug 27 2024

Mathematica

seq[max_] := Module[{ps = Select[Range[Floor[Surd[max, 8]]], PrimeQ], t}, t = Select[Tuples[ps, {3}], UnsameQ @@ # &]; t = Select[t, #[[1]] < Surd[max, (#[[2]]^#[[3]])] &]; Sort[Select[(#[[1]]^(#[[2]]^#[[3]])) & /@ t, # <= max &]]]; seq[10^14] (* Amiram Eldar, Aug 27 2024 *)

Crossrefs

Subsequence of A217709.

Sequence in context: A164987 A176765 A290499 * A016820 A016856 A016976

Adjacent sequences: A375730 A375731 A375732 * A375734 A375735 A375736

Keyword

easy,nonn,new

Author

Paul Duckett, Aug 25 2024

Status

approved