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A124142
Abundant numbers k such that sigma(k) is a perfect power.
1
66, 70, 102, 210, 282, 364, 400, 510, 642, 690, 714, 770, 820, 930, 966, 1080, 1092, 1146, 1164, 1200, 1416, 1566, 1624, 1672, 1782, 2130, 2226, 2250, 2346, 2460, 2530, 2586, 2652, 2860, 2910, 2912, 3012, 3198, 3210, 3340, 3498, 3522, 3560, 3710, 3810
OFFSET
1,1
COMMENTS
Positive integers k such that sigma(k) > 2*k and sigma(k) = a^b where both a and b are greater than 1.
If k is a term with sigma(k) a square, and p and q are members of A066436 that do not divide k, then k*p*q is in the sequence. Thus if A066436 is infinite, so is this sequence. - Robert Israel, Oct 29 2018
LINKS
EXAMPLE
a(1) = 66 since sigma(66) = 144 = 12^2.
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:=[]: 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]; fi od od;
MATHEMATICA
filterQ[n_] := With[{s = DivisorSigma[1, n]}, s > 2n && GCD @@ FactorInteger[s][[All, 2]] > 1];
Select[Range[4000], filterQ] (* Jean-François Alcover, Sep 16 2020 *)
PROG
(PARI) is(k) = {my(s = sigma(k)); s > 2*k && ispower(s); } \\ Amiram Eldar, Aug 02 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Walter Kehowski, Dec 01 2006
STATUS
approved