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A306796 Primitive abundant numbers (A071395) that are squares.

%I A306796 #21 Mar 12 2025 04:45:03
%S A306796 342225,280495504,1029447225,1148667664,1435045924,1596961444,
%T A306796 1757705625,2177622225,14787776025,18114198921,32871503025,
%U A306796 45018230625,150897287025,245485566225,272993710144,296006724225,705373218225,1126920249225,1329226832241,1358425215225
%N A306796 Primitive abundant numbers (A071395) that are squares.
%C A306796 The square roots of the terms are 585, 16748, 32085, 33892, 37882, 39962, 41925, 46665, 121605, 134589, ...
%H A306796 Amiram Eldar, <a href="/A306796/b306796.txt">Table of n, a(n) for n = 1..5400</a>
%t A306796 abQ[f_] := Times@@((f[[;;,1]]^(f[[;;,2]]+1)-1)/(f[[;;,1]]-1)) > 2*Times@@Power@@@f;
%t A306796 nondefQ[f_,g_] := Times@@((f^(g+1)-1)/(f-1)) >= 2*Times@@(f^g);
%t A306796 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 A306796 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 A306796 papowerQ[n_, e_] := Module[{f=FactorInteger[n]}, f[[;;,2]]*=e; paQ[f]];
%t A306796 s={}; e=2; Do[If[papowerQ[m, e], AppendTo[s, m^e]], {m, 2, 50000}]; s
%o A306796 (PARI) is1(k) = {my(f = factor(k)); for(i = 1, #f~, f[i, 2] *= 2); 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 A306796 list(lim) = for(k = 1, lim, if(is1(k), print1(k^2, ", "))); \\ _Amiram Eldar_, Mar 12 2025
%Y A306796 Intersection of A000290 and A071395.
%Y A306796 Cf. A306797, A379949 (most likely gives the odd terms).
%K A306796 nonn
%O A306796 1,1
%A A306796 _Amiram Eldar_, Mar 10 2019