A258106 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.

1998, 3876, 4524, 10062, 21582, 45220, 52780, 85428, 125976, 226100, 263900, 271092, 511428, 597012, 602946, 839106, 1033974, 1130500, 1274724, 1280532, 1319500, 1435764, 1469720, 1575860, 1810926, 1895706, 2171364, 2550636, 3162740, 4083366, 4766034, 5652500

Offset

1,1

Comments

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.

Links

Amiram Eldar, Table of n, a(n) for n = 1..1000

Example

usigma(1998) = 3192, (-1)sigma(1998) = 1368 and 3191 + 1368 = 4560 = sigma(1998);
usigma(3876) = 7200, (-1)sigma(3876) = 2880 and 7200 + 2880 = 10080 = sigma(3876);
usigma(4524) = 8400, (-1)sigma(4524) = 3360 and 8400 + 3360 = 11760 = sigma(4524); etc.

Maple

with(numtheory): P:=proc(q) local a, b, c, d, i, k, n; a:=0; b:=0;
for n from 1 to q do a:=divisors(n); d:=0; for k from 1 to nops(a) do
if gcd(a[k], n/a[k])>1 then d:=d+a[k]; fi; od; a:=ifactors(n)[2]; b:=1;
for i from 1 to nops(a) do b:=b*(-1+sum(a[i][1]^j, j=1..a[i][2])); od;
if b=d then print(n); fi; od; end: P(10^9);

Mathematica

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 *)

Crossrefs

Cf. A000203, A034448, A049060, A258101.

Sequence in context: A063054 A296826 A171467 * A253744 A253751 A031808

Adjacent sequences: A258103 A258104 A258105 * A258107 A258108 A258109

Keyword

nonn

Author

Paolo P. Lava, May 20 2015

Extensions

More terms from Amiram Eldar, Jun 25 2019

Status

approved