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A258101 Number x such that usigma(x) = (-1)sigma(x), where usigma(x) is the sum of unitary divisors of x (A034448) and (-1)sigma(x) is defined in A049060 .

%I A258101 #16 Jun 23 2019 02:59:56
%S A258101 1,4,15867,21357,49887,63468,69875,85428,86387,149875,199548,247475,
%T A258101 271607,279500,293944,318681,345548,599500,637659,989900,1086428,
%U A258101 1169091,1274724,1897875,1913571,2550636,2665269,2801880,2855691
%N A258101 Number x such that usigma(x) = (-1)sigma(x), where usigma(x) is the sum of unitary divisors of x (A034448) and (-1)sigma(x) is defined in A049060 .
%H A258101 Amiram Eldar, <a href="/A258101/b258101.txt">Table of n, a(n) for n = 1..1000</a>
%e A258101 usigma(1) = (-1)sigma(1) = 1;
%e A258101 usigma(4) =  (-1)sigma(4) = 5;
%e A258101 usigma(15867) = (-1)sigma(15867) = 18480; etc.
%p A258101 with(numtheory): P:=proc(q) local a,b,d,k,n;
%p A258101 a:=0; for n from 1 to q do a:=divisors(n); d:=0; for k from 1 to nops(a)
%p A258101 do if gcd(a[k],n/a[k])=1 then d:=d+a[k]; fi; od; a:=ifactors(n)[2]; b:=1;
%p A258101 for k from 1 to nops(a) do b:=b*(-1+sum(a[k][1]^j,j=1..a[k][2])); od;
%p A258101 if b=d then print(n); fi; od; end: P(10^9);
%t A258101 aQ[n_] := Module[{f = FactorInteger[n]}, p = f[[;;,1]]; e = f[[;;,2]]; Times@@(p^e+1) == Times@@((p^(e+1)-2*p+1)/(p-1))]; Join[{1}, Select[Range[2, 200000 ], aQ]] (* _Amiram Eldar_, Jun 23 2019 *)
%Y A258101 Cf. A034448, A049060, A258106.
%K A258101 nonn
%O A258101 1,2
%A A258101 _Paolo P. Lava_, May 20 2015