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%I #6 Mar 13 2019 23:51:37
%S 6886512413632368153,8815747507513708671,334845050584968548307656,
%T 1254177078562232856388071,27869863573964698956703125
%N Primitive abundant numbers (A071395) that are cubes.
%C The cube roots of the terms are 1902537, 2065791, 69440786, 107841591, 303187725, ...
%t abQ[f_] := Times@@((f[[;;,1]]^(f[[;;,2]]+1)-1)/(f[[;;,1]]-1)) > 2*Times@@Power@@@f;
%t nondefQ[f_,g_] := Times@@((f^(g+1)-1)/(f-1)) >= 2*Times@@(f^g);
%t sub[f_,k_] := Module[{g=f[[;;,2]]}, n=Length[g]; kk=k-1; Do[g[[i]] = Mod[kk, f[[i,2]]+1]; kk=(kk-g[[i]])/(f[[i,2]]+1), {i,1,n}]; g];
%t paQ[f_] := abQ[f] && Module[{nd = Times@@(f[[;;,2]]+1), ans=True}, Do[g=sub[f,k]; If[nondefQ[f[[;;,1]], g], ans=False; Break[]], {k,1,nd-1}]; ans];
%t papowerQ[n_, e_] := Module[{f=FactorInteger[n]}, f[[;;,2]]*=e; paQ[f]];
%t s={}; e=3; Do[If[papowerQ[m, e], AppendTo[s, m^e]], {m, 2, 7*10^7}]; s
%Y Cf. A000578, A071395, A306796.
%K nonn,more
%O 1,1
%A _Amiram Eldar_, Mar 10 2019
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