A124142 Abundant numbers k such that sigma(k) is a perfect power.

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

Author

Walter Kehowski, Dec 01 2006

Keywords

• nonn

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

Examples

			a(1) = 66 since sigma(66) = 144 = 12^2.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001597, A005101, A065496, A066436.

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:=[]: 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 *)

PARI

is(k) = {my(s = sigma(k)); s > 2*k && ispower(s);} \\ Amiram Eldar, Aug 02 2024