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
Keywords
Examples
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.
Links
- Amiram Eldar, Table of n, a(n) for n = 1..1000
Programs
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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);
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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 *)
Extensions
More terms from Amiram Eldar, Jun 25 2019
Comments