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%I #59 Mar 29 2024 10:22:22
%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
%C In particular, these numbers have abundancy 2 to 3: 2 < sigma(n)/n <= 3. - _Charles R Greathouse IV_, Jan 30 2014
%C Subsequence of A083207. - _Ivan N. Ianakiev_, Mar 20 2017
%C The concept of admirable numbers was developed by educator Jerome Michael Sachs (1914-2012) for a television in-service training course in mathematics for elementary school teachers. - _Amiram Eldar_, Aug 22 2018
%C Odd terms are listed in A109729. For abundant nonsquares, it is equivalent to say sigma(n)/2 - n divides n. For squares, sigma(n)/2 - n is half-integer, but n could still be an integer multiple. This first occurs for n = m^2 with even m = 2^k*(2^(2*k+1)-1), k = 1, 2, 3, 6, ... (A146768), and odd m = 13167. - _M. F. Hasler_, Jan 26 2020
%H Charles R Greathouse IV, <a href="/A111592/b111592.txt">Table of n, a(n) for n = 1..10000</a>
%H F. Firoozbakht, M. F. Hasler, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Hasler/hasler2.html">Variations on Euclid's formula for Perfect Numbers</a>, JIS 13 (2010) #10.3.1.
%H Giovanni Resta, <a href="http://www.numbersaplenty.com/set/admirable_number/">admirable numbers</a>
%H J. M. Sachs, <a href="https://www.jstor.org/stable/41184328">Admirable Numbers and Compatible Pairs</a>, The Arithmetic Teacher, Vol. 7, No. 6 (1960), pp. 293-295.
%H T. Trotter, <a href="https://web.archive.org/web/20101130221929/http://www.trottermath.net/numthry/admirabl.html">Admirable Numbers</a> [Warning: As of March 2018 this site appears to have been hacked. Proceed with great caution. The original content should be retrieved from the Wayback machine and added here. - _N. J. A. Sloane_, Mar 29 2018]
%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[ # ] &] (* _Robert G. Wilson v_, Aug 13 2005 *)
%t Select[Range[812],MemberQ[Most[Divisors[#]],(DivisorSigma[1,#]-2*#)/2]&] (* _Ivan N. Ianakiev_, Mar 23 2017 *)
%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 Subsequence of A005101 (abundant numbers).
%Y Cf. A000396 (perfect numbers), A005100 (deficient numbers), A000203 (sigma), A061645.
%Y Cf. A109729 (odd admirable numbers).
%K easy,nonn
%O 1,1
%A _Jason Earls_, Aug 09 2005
%E Better definition from _Walter Kehowski_, Aug 12 2005
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