A306796 Primitive abundant numbers (A071395) that are squares.

342225, 280495504, 1029447225, 1148667664, 1435045924, 1596961444, 1757705625, 2177622225, 14787776025, 18114198921, 32871503025, 45018230625, 150897287025, 245485566225, 272993710144, 296006724225, 705373218225, 1126920249225, 1329226832241, 1358425215225

Offset

1,1

Comments

The square roots of the terms are 585, 16748, 32085, 33892, 37882, 39962, 41925, 46665, 121605, 134589, ...

Links

Amiram Eldar, Table of n, a(n) for n = 1..5400

Mathematica

abQ[f_] := Times@@((f[[;; , 1]]^(f[[;; , 2]]+1)-1)/(f[[;; , 1]]-1)) > 2*Times@@Power@@@f;
nondefQ[f_, g_] := Times@@((f^(g+1)-1)/(f-1)) >= 2*Times@@(f^g);
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];
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];
papowerQ[n_, e_] := Module[{f=FactorInteger[n]}, f[[;; , 2]]*=e; paQ[f]];
s={}; e=2; Do[If[papowerQ[m, e], AppendTo[s, m^e]], {m, 2, 50000}]; s

Prog

(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; }
list(lim) = for(k = 1, lim, if(is1(k), print1(k^2, ", "))); \\ Amiram Eldar, Mar 12 2025

Crossrefs

Intersection of A000290 and A071395.

Cf. A306797, A379949 (most likely gives the odd terms).

Sequence in context: A266171 A233604 A363175 * A379949 A365611 A029568

Adjacent sequences: A306793 A306794 A306795 * A306797 A306798 A306799

Keyword

nonn,changed

Author

Amiram Eldar, Mar 10 2019

Status

approved