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%I
%S 12,20,24,30,40,42,54,56,66,70,78,84,88,102,104,114,120,138,140,174,
%T 186,222,224,234,246,258,270,282,308,318,354,364,366,368,402,426,438,
%U 464,474,476,498,532,534,582,606,618,642,644,650,654,672,678,762,786,812
%N Admirable numbers. A number n is admirable if there exists a proper divisor d' of n such that sigma(n)-2d'=2n, where sigma(n) is the sum of all divisors of n.
%C All admirable numbers are abundant.
%C If 2^n-2^k-1 is an odd prime then m=2^(n-1)*(2^n-2^k-1) is in the sequence because 2^k is one of the proper divisors of m and sigma(m)-2m=(2^n-1)*(2^n-2^k)-2^n*(2^n-2^k-1)=2^k hence m=(sigma(m)-m)-2^k, namely m is an Admirable number. This is one of the results of the following theorem that I have found. Theorem: If 2^n-j-1 is an odd prime and m=2^(n-1)*(2^n-j-1) then sigma(m)-2m=j. The case j=0 is well known. - _Farideh Firoozbakht_, Jan 28 2006
%H T. Trotter, <a href="http://www.trottermath.net/numthry/admirabl.html">Admirable Numbers</a>
%H Charles R Greathouse IV, <a href="/A111592/b111592.txt">Table of n, a(n) for n = 1..10000</a>
%e 12 = 1+3+4+6-2, 20 = 2+4+5+10-1, etc.
%p with(numtheory); isadmirable := proc(n) local b, d, S; b:=false; S:=divisors(n) minus {n}; for d in S do if sigma(n)-2*d=2*n then b:=true; break fi od; return b; end: select(proc(z) isadmirable(z) end, [$1..1000]); (_Walter Kehowski_, Aug 12 2005)
%t fQ[n_] := Block[{d = Most[Divisors[n]], k = 1}, l = Length[d]; s = Plus @@ d; While[k < l && s - 2d[[k]] > n, k++ ]; If[k > l || s != n + 2d[[k]], False, True]]; Select[ Range[821], fQ[ # ] &] (from _Robert G. Wilson v_, Aug 13 2005)
%o (PARI) for(n=1,10^3,ap=sigma(n)-2*n;if(ap>0 && (ap%2)==0,d=ap/2;if(d!=n && (n%d)==0, print1(n",")))) - Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 30 2008
%o (PARI) is(n)=if(issquare(n)||issquare(n/2),0,my(d=sigma(n)/2-n); d>0 && d!=n && n%d==0) \\ _Charles R Greathouse IV_, Jun 21 2011
%Y Cf. A000396, A005101, A005100, A000203, A061645.
%K easy,nonn,changed
%O 1,1
%A Jason Earls (zevi_35711(AT)yahoo.com), Aug 09 2005
%E Better definition from _Walter Kehowski_, Aug 12 2005
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