A258106 - OEIS

%I #15 Jun 25 2019 03:11:13

%S 1998,3876,4524,10062,21582,45220,52780,85428,125976,226100,263900,

%T 271092,511428,597012,602946,839106,1033974,1130500,1274724,1280532,

%U 1319500,1435764,1469720,1575860,1810926,1895706,2171364,2550636,3162740,4083366,4766034,5652500

%N Number x such that sigma(x) = usigma(x) + (-1)sigma(x), where sigma(x) is the sum of divisors of x (A000203), usigma(x) is the sum of unitary divisors of x (A034448) and (-1)sigma(x) is defined in A049060.

%C The definition implies that the terms of the sequence could be defined as the numbers x such that (-1)sigma(x) is equal to the sum of the non-unitary divisors of x.

%H Amiram Eldar, <a href="/A258106/b258106.txt">Table of n, a(n) for n = 1..1000</a>

%e usigma(1998) = 3192, (-1)sigma(1998) = 1368 and 3191 + 1368 = 4560 = sigma(1998);

%e usigma(3876) = 7200, (-1)sigma(3876) = 2880 and 7200 + 2880 = 10080 = sigma(3876);

%e usigma(4524) = 8400, (-1)sigma(4524) = 3360 and 8400 + 3360 = 11760 = sigma(4524); etc.

%p with(numtheory): P:=proc(q) local a,b,c,d,i,k,n; a:=0; b:=0;

%p for n from 1 to q do a:=divisors(n); d:=0; for k from 1 to nops(a) do

%p if gcd(a[k],n/a[k])>1 then d:=d+a[k]; fi; od; a:=ifactors(n)[2]; b:=1;

%p for i from 1 to nops(a) do b:=b*(-1+sum(a[i][1]^j,j=1..a[i][2])); od;

%p if b=d then print(n); fi; od; end: P(10^9);

%t aQ[n_] := Module[{f = FactorInteger[n]}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times@@((p^(e+1)-1)/(p-1)) == Times@@(p^e+1) + Times@@((p^(e+1)-2*p+1)/(p-1))]; Select[Range[2, 100000], aQ] (* _Amiram Eldar_, Jun 25 2019 *)

%Y Cf. A000203, A034448, A049060, A258101.

%K nonn

%O 1,1

%A _Paolo P. Lava_, May 20 2015

%E More terms from _Amiram Eldar_, Jun 25 2019