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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. 26
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 (list; graph; refs; listen; history; text; internal format)
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

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

T. Trotter, Admirable Numbers

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[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

CROSSREFS

Subsequence of A005101.

Cf. A000396, A005100, A000203, A061645.

Sequence in context: A112769 A097320 A204825 * A111947 A109396 A286004

Adjacent sequences:  A111589 A111590 A111591 * A111593 A111594 A111595

KEYWORD

easy,nonn

AUTHOR

Jason Earls (zevi_35711(AT)yahoo.com), Aug 09 2005

EXTENSIONS

Better definition from Walter Kehowski, Aug 12 2005

STATUS

approved

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Last modified July 26 16:42 EDT 2017. Contains 289839 sequences.