A190665 Numbers k such that sum of aliquot parts of k is a nontrivial power: sigma(k) - k = a^b for integers a>1, b>1.

9, 10, 12, 15, 24, 26, 49, 56, 58, 69, 75, 76, 90, 95, 119, 122, 124, 133, 140, 143, 147, 153, 176, 194, 215, 243, 287, 332, 363, 386, 407, 429, 477, 495, 507, 511, 524, 527, 536, 551, 568, 575, 578, 688, 717, 738, 791, 794, 815, 867, 871, 892, 924, 935, 961, 963, 992, 1001, 1018, 1075, 1083, 1159, 1196, 1199, 1243, 1295, 1304, 1324, 1346, 1391, 1415, 1421, 1431, 1532, 1573, 1587

Offset: 1

Author

Antonio Roldán, May 16 2011

Keywords

• nonn

Comments

Similar to A065496.

Examples

			122: aliquot parts: 1, 2, 61, sum: 1+2+61 = 64 = 8^2
140: sum of aliquot parts: 1+2+4+5+7+10+14+20+28+35+70 = 196 = 14^2.
		

Crossrefs

Cf. A065496, A001597, A048699.

Programs

Maple

isA001597 := proc(n) for a from 2 do if a^2 > n then return false; end if; for b from 2 do if a^b =n then return true; elif a^b>n then break; end if; end do; end do: end proc:
isA190665 := proc(n) isA001597(numtheory[sigma](n)-n) ; end proc:
for n from 1 to 2000 do if isA190665(n) then printf("%d,",n) ; end if; end do; # R. J. Mathar, May 30 2011

Mathematica

powerQ[n_] := GCD @@ FactorInteger[n][[All, 2]] > 1;
okQ[n_] := powerQ[DivisorSigma[1, n] - n];
Select[Range[2000], okQ] (* Jean-François Alcover, Jul 31 2024 *)

PARI

ypower(n)= { local(f, p=0); f=factor(n); if(gcd(f[, 2])>1,p=1); return(p) }
{  for (n=1, 1000, a=sigma(n)-n; if(ypower(a), print1(n," "))) }
/* Antonio Roldán, Oct 23 2012 */