A111592 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.

12, 20, 24, 30, 40, 42, 54, 56, 66, 70, 78, 84, 88, 102, 104, 114, 120, 138, 140, 174, 186, 222, 224, 234, 246, 258, 270, 282, 308, 318, 354, 364, 366, 368, 402, 426, 438, 464, 474, 476, 498, 532, 534, 582, 606, 618, 642, 644, 650, 654, 672, 678, 762, 786, 812

Offset: 1

Author

Jason Earls, Aug 09 2005

Keywords

• easy
• nonn

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

Examples

			12 = 1+3+4+6-2, 20 = 2+4+5+10-1, etc.
		

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

Crossrefs

Subsequence of A005101 (abundant numbers).

Cf. A000396 (perfect numbers), A005100 (deficient numbers), A000203 (sigma), A061645.

Cf. A109729 (odd admirable numbers).

Programs

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 *)

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

Extensions

Better definition from Walter Kehowski, Aug 12 2005