A124144 Perfect powers pp such that sigma(k) = pp for some abundant number k.

144, 216, 576, 784, 961, 1296, 1728, 1764, 2304, 2744, 3136, 3600, 3844, 4356, 5184, 6084, 7056, 7776, 8100, 9216, 11664, 12544, 13824, 14400, 15376, 15876, 17424, 19600, 20736, 21952, 24336, 27000, 28224, 32400, 34596, 36864, 38416, 39204, 41616, 44100, 46656, 50176

Offset: 1

Author

Walter Kehowski, Dec 01 2006

Keywords

• nonn

Examples

			a(1) = 144 since sigma(66) = 144 > 2*66 = 132.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001597, A005101, A065496.

Programs

Maple

with(numtheory); egcd := proc(n::posint) local L; if n>1 then L:=ifactors(n)[2]; L:=map(z->z[2],L); return igcd(op(L)) else return 1 fi; end; L:=[]: P:={}: for w to 1 do for n from 1 to 10000 do s:=sigma(n); if s>2*n and egcd(s)>1 then print(n,s,ifactor(s)); L:=[op(L),n]; P:=P union {s}; fi od od; L; P;

Mathematica

ppQ[n_] := GCD @@ FactorInteger[n][[;; , 2]] > 1;
f[n_] := Module[{s = DivisorSigma[1, n]}, If[s > 2*n, s, Nothing]];
seq[max_] := Union[Select[Array[f, max], # < max && ppQ[#] &]]; seq[60000] (* Amiram Eldar, Mar 11 2024 *)

Extensions

a(32) inserted and more terms added by Amiram Eldar, Mar 11 2024