

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.


24



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
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OFFSET

1,1


COMMENTS

All admirable numbers are abundant.
If 2^n2^k1 is an odd prime then m=2^(n1)*(2^n2^k1) is in the sequence because 2^k is one of the proper divisors of m and sigma(m)2m=(2^n1)*(2^n2^k)2^n*(2^n2^k1)=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^nj1 is an odd prime and m=2^(n1)*(2^nj1) 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


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+62, 20 = 2+4+5+101, 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 *)


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)/2n); 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 A055598
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



