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A306797 Primitive abundant numbers (A071395) that are cubes.

%I A306797 #15 Mar 12 2025 04:45:10
%S A306797 6886512413632368153,8815747507513708671,334845050584968548307656,
%T A306797 1254177078562232856388071,27869863573964698956703125,
%U A306797 108182814324640834480192546875,384852900473651366592567235048,520616176957487045802123463832,567962434462802770687173681448,1389387861291307410644039382069
%N A306797 Primitive abundant numbers (A071395) that are cubes.
%C A306797 The cube roots of the terms are 1902537, 2065791, 69440786, 107841591, 303187725, ...
%H A306797 Amiram Eldar, <a href="/A306797/b306797.txt">Table of n, a(n) for n = 1..24</a>
%t A306797 abQ[f_] := Times@@((f[[;;,1]]^(f[[;;,2]]+1)-1)/(f[[;;,1]]-1)) > 2*Times@@Power@@@f;
%t A306797 nondefQ[f_,g_] := Times@@((f^(g+1)-1)/(f-1)) >= 2*Times@@(f^g);
%t A306797 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 A306797 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 A306797 papowerQ[n_, e_] := Module[{f=FactorInteger[n]}, f[[;;,2]]*=e; paQ[f]];
%t A306797 s={}; e=3; Do[If[papowerQ[m, e], AppendTo[s, m^e]], {m, 2, 7*10^7}]; s
%o A306797 (PARI) is1(k) = {my(f = factor(k)); for(i = 1, #f~, f[i, 2] *= 3); if(sigma(f, -1) <= 2, return(0)); for(i = 1, #f~, f[i, 2] -= 1; if(sigma(f, -1) >= 2, return(0)); f[i, 2] += 1); 1;}
%o A306797 list(lim) = for(k = 1, lim, if(is1(k), print1(k^3, ", "))); \\ _Amiram Eldar_, Mar 12 2025
%Y A306797 Intersection of A000578 and A071395.
%Y A306797 Cf. A306796.
%K A306797 nonn
%O A306797 1,1
%A A306797 _Amiram Eldar_, Mar 10 2019
%E A306797 a(6)-a(10) from _Amiram Eldar_, Mar 12 2025