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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.

%I A190665 #24 Jul 31 2024 09:06:55
%S A190665 9,10,12,15,24,26,49,56,58,69,75,76,90,95,119,122,124,133,140,143,147,
%T A190665 153,176,194,215,243,287,332,363,386,407,429,477,495,507,511,524,527,
%U A190665 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
%N 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.
%C A190665 Similar to A065496.
%e A190665 122: aliquot parts: 1, 2, 61, sum: 1+2+61 = 64 = 8^2
%e A190665 140: sum of aliquot parts: 1+2+4+5+7+10+14+20+28+35+70 = 196 = 14^2.
%p A190665 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:
%p A190665 isA190665 := proc(n) isA001597(numtheory[sigma](n)-n) ; end proc:
%p A190665 for n from 1 to 2000 do if isA190665(n) then printf("%d,",n) ; end if; end do; # _R. J. Mathar_, May 30 2011
%t A190665 powerQ[n_] := GCD @@ FactorInteger[n][[All, 2]] > 1;
%t A190665 okQ[n_] := powerQ[DivisorSigma[1, n] - n];
%t A190665 Select[Range[2000], okQ] (* _Jean-François Alcover_, Jul 31 2024 *)
%o A190665 (PARI)
%o A190665 ypower(n)= { local(f, p=0); f=factor(n); if(gcd(f[, 2])>1,p=1); return(p) }
%o A190665 {  for (n=1, 1000, a=sigma(n)-n; if(ypower(a), print1(n," "))) }
%o A190665 /* _Antonio Roldán_, Oct 23 2012 */
%Y A190665 Cf. A065496, A001597, A048699.
%K A190665 nonn
%O A190665 1,1
%A A190665 _Antonio Roldán_, May 16 2011