A379949
Primitive abundant numbers (A091191) that are odd squares.
Original entry on oeis.org
342225, 1029447225, 1757705625, 2177622225, 14787776025, 18114198921, 32871503025, 45018230625, 150897287025, 245485566225, 296006724225, 705373218225, 1126920249225, 1329226832241, 1358425215225, 1545732725625, 1555265892609, 1783322538921, 2811755495241, 4627123655625, 5248080775161, 6140855705625, 7683069267225
Offset: 1
Subsequences of the following sequences:
A306796 (odd terms, but only if there are no odd perfect numbers),
A363176,
A379503 (conjectured).
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is_A379949(n) = if(!(n%2) || !issquare(n) || sigma(n)<=2*n, 0, fordiv(n, d, if(d>1 && sigma(n/d, -1)>2, return(0))); (1));
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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) = forstep(k = 1, lim, 2, if(is1(k), print1(k^2, ", "))); \\ Amiram Eldar, Mar 12 2025
A306797
Primitive abundant numbers (A071395) that are cubes.
Original entry on oeis.org
6886512413632368153, 8815747507513708671, 334845050584968548307656, 1254177078562232856388071, 27869863573964698956703125, 108182814324640834480192546875, 384852900473651366592567235048, 520616176957487045802123463832, 567962434462802770687173681448, 1389387861291307410644039382069
Offset: 1
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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=3; Do[If[papowerQ[m, e], AppendTo[s, m^e]], {m, 2, 7*10^7}]; s
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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;}
list(lim) = for(k = 1, lim, if(is1(k), print1(k^3, ", "))); \\ Amiram Eldar, Mar 12 2025
A363175
Primitive abundant numbers (A071395) that are powerful numbers (A001694).
Original entry on oeis.org
342225, 570375, 3172468, 4636684, 63126063, 99198099, 117234117, 171991125, 280495504, 319600125, 327921075, 404529741, 581549787, 635689593, 762155163, 1029447225, 1148667664, 1356949503, 1435045924, 1501500375, 1558495125, 1596961444, 1757705625, 1778362047
Offset: 1
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f1[p_, e_] := (p^(e + 1) - 1)/(p^(e + 1) - p^e); f2[p_, e_] := (p^(e + 1) - p)/(p^(e + 1) - 1);
primAbQ[n_] := (r = Times @@ f1 @@@ (f = FactorInteger[n])) > 2 && r * Max @@ f2 @@@ f < 2;
seq[max_] := Module[{pow = Union[Flatten[Table[i^2*j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}]]]}, Select[Rest[pow], primAbQ]]; seq[10^10]
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isPrimAb(n) = {my(f = factor(n), r, p, e); r = sigma(f, -1); r > 2 && vecmax(vector(#f~, i, p = f[i, 1]; e = f[i, 2]; (p^(e + 1) - p)/(p^(e + 1) - 1))) * r < 2; }
lista(lim) = {my(pow = List(), t); for(j=1, sqrtnint(lim\1, 3), for(i=1, sqrtint(lim\j^3), listput(pow, i^2*j^3))); select(x->isPrimAb(x), Set(pow)); }
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