OFFSET
1,1
COMMENTS
All admirable numbers are abundant.
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
In particular, these numbers have abundancy 2 to 3: 2 < sigma(n)/n <= 3. - Charles R Greathouse IV, Jan 30 2014
Subsequence of A083207. - Ivan N. Ianakiev, Mar 20 2017
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
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
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
F. Firoozbakht, M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.
Giovanni Resta, admirable numbers
J. M. Sachs, Admirable Numbers and Compatible Pairs, The Arithmetic Teacher, Vol. 7, No. 6 (1960), pp. 293-295.
T. Trotter, Admirable Numbers [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]
EXAMPLE
12 = 1+3+4+6-2, 20 = 2+4+5+10-1, etc.
MAPLE
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
MATHEMATICA
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 *)
Select[Range[812], MemberQ[Most[Divisors[#]], (DivisorSigma[1, #]-2*#)/2]&] (* Ivan N. Ianakiev, Mar 23 2017 *)
PROG
(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
(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
KEYWORD
easy,nonn
AUTHOR
Jason Earls, Aug 09 2005
EXTENSIONS
Better definition from Walter Kehowski, Aug 12 2005
STATUS
approved