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A124144
Perfect powers pp such that sigma(k) = pp for some abundant number k.
1
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
1,1
LINKS
EXAMPLE
a(1) = 144 since sigma(66) = 144 > 2*66 = 132.
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Walter Kehowski, Dec 01 2006
EXTENSIONS
a(32) inserted and more terms added by Amiram Eldar, Mar 11 2024
STATUS
approved