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

%I A375733 #16 Oct 01 2024 15:51:44
%S A375733 390625,1953125,5764801,40353607,214358881,815730721,2357947691,
%T A375733 6975757441,10604499373,16983563041,78310985281,118587876497,
%U A375733 322687697779,500246412961,847288609443,852891037441,1801152661463,3512479453921,7984925229121,11688200277601,14507145975869
%N A375733 Integers of the form p^(q^r) where p, q, r are distinct primes.
%C A375733 a(15) is the first term not an eighth or ninth power.
%e A375733 390625 is a term because it equals 5^(2^3).
%e A375733 118587876497 is a term because it equals 17^(3^2).
%p A375733 N:= 10^20: # for terms <= N
%p A375733 with(priqueue):
%p A375733 initialize(pq):
%p A375733 if 2^(3^5) < N then insert([-2^(3^5),2,3,5],pq) fi:
%p A375733 if 2^(5^3) < N then insert([-2^(5^3),2,5,3],pq) fi:
%p A375733 if 3^(2^5) < N then insert([-3^(2^5),3,2,5],pq) fi:
%p A375733 if 3^(5^2) < N then insert([-3^(5^2),3,5,2],pq) fi:
%p A375733 if 5^(2^3) < N then insert([-5^(2^3),5,2,3],pq) fi:
%p A375733 if 5^(3^2) < N then insert([-5^(3^2),5,3,2],pq) fi:
%p A375733 Res:= NULL:
%p A375733 while pq[0] > 0 do
%p A375733   t:= extract(pq);
%p A375733   Res:= Res, -t[1];
%p A375733   q:= nextprime(t[2]);
%p A375733   while member(q,{t[3],t[4]}) do q:= nextprime(q) od;
%p A375733   v:= q^(t[3]^t[4]);
%p A375733   if v <= N then insert([-v,q,t[3],t[4]],pq) fi;
%p A375733   if t[2] = 2 then
%p A375733     q:= nextprime(t[3]);
%p A375733     if q = t[4] then q:= nextprime(q) fi;
%p A375733     v:= 2^(q^t[4]);
%p A375733     if v <= N then insert([-v,2,q,t[4]],pq) fi;
%p A375733   fi;
%p A375733   if {t[2],t[3]} = {2,3} then
%p A375733     q:= nextprime(t[4]);
%p A375733     v:= t[2]^(t[3]^q);
%p A375733     if v <= N then insert([-v,t[2],t[3],q],pq) fi;
%p A375733   fi
%p A375733 od:
%p A375733 Res; # _Robert Israel_, Aug 27 2024
%t A375733 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 *)
%Y A375733 Subsequence of A217709.
%K A375733 easy,nonn
%O A375733 1,1
%A A375733 _Paul Duckett_, Aug 25 2024