A034094 - OEIS

%I #22 Jul 07 2022 08:11:46

%S 1,20,312,9744,29280,53352,1666224,5006880,106798080,133301760,

%T 980733600,9099742080,18262471680,22794600960,1556055895680,

%U 3577201689600,4464942451200,380428773854896765462278360268800000

%N (-1)sigma perfect numbers: (-1)sigma(a) = m*a for some integer m, where if a = Product p(i)^r(i) then (-1)sigma(a) = Product_{i} (-1 + Sum_{s=1..r(i)} p(i)^s).

%C The indices of some terms are 1, so these numbers are fixed points of (-1)sigma where (-1)sigma is A049060.

%e Factorizations 2^2*5, 2^3*3*13, 2^4*3*7*29, 2^5*3*5*61, 2^3*3^3*13*19, 2^4*3^3*7*19*29, 2^5*3^3*5*19*61, 2^10*3*5*17*409, 2^9*3*5*17*1021, 2^5*3^2*5^2*7*11*29*61, 2^7*3*5*11^2*13*23*131, 2^10*3^3*5*17*19*409, 2^9*3^3*5*17*19*1021, 2^7*3^3*5*11^2*13*19*23*131, 2^10*3^2*5^2*7*11*17*29*409, 2^9*3^2*5^2*7*11*17*29*1021, 2^24*3^3*5^5*7^2*11*17*19*29*61*233*239*467*479*70051.

%t f[p_, e_] := (p^(e+1)-2*p+1)/(p-1); s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[10^5], Divisible[s[#], #] &] (* _Amiram Eldar_, Jul 07 2022 *)

%o (PARI) msig(n) = {f = factor(n); for (i=1, #f~, f[i, 1] = (f[i,1]^(f[i,2]+1)-2*f[i,1]+1)/(f[i,1]-1); f[i, 2] = 1;); factorback(f);}

%o isok(n) = denominator(msig(n)/n) == 1; \\ _Michel Marcus_, Jun 02 2016

%Y Cf. A034095, A049060.

%K nonn,more

%O 1,2

%A _Yasutoshi Kohmoto_

%E a(1)=1 prepended by _Michel Marcus_, Jun 02 2016

%E a(10) and a(11) switched and missing term a(13) inserted by _Amiram Eldar_, Jul 07 2022