%I A006037 M5339
%S A006037 70,836,4030,5830,7192,7912,9272,10430,10570,10792,10990,11410,11690,
%T A006037 12110,12530,12670,13370,13510,13790,13930,14770,15610,15890,16030,
%U A006037 16310,16730,16870,17272,17570,17990,18410,18830,18970,19390,19670
%N A006037 Weird numbers: abundant (A005101) but not pseudoperfect (A005835).
%C A006037 There are no odd weird numbers < 10^17. - Robert A. Hearn (rah(AT)ai.mit.edu), 
               May 25 2005
%C A006037 Contribution from Alois P. Heinz (heinz(AT)hs-heilbronn.de), Oct 30 2009: 
               (Start)
%C A006037 The first weird number that has more than one decomposition of their 
               divisors set into two subsets with equal sum (and thus is not a member 
               of A083209) is 10430:
%C A006037 1+5+7+10+14+35+298+10430 = 2+70+149+745+1043+1490+2086+5215
%C A006037 2+70+298+10430 = 1+5+7+10+14+35+149+745+1043+1490+2086+5215 (End)
%D A006037 S. Benkoski, "Are All Weird Numbers Even?", Problem E2308, Amer. Math. 
               Monthly, 79 (1972), 774.
%D A006037 S. J. Benkoski and P. Erdos, On weird and pseudoperfect numbers, Math. 
               Comp., 28 (1974), 617-623.
%D A006037 J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 70, p. 24, Ellipses, 
               Paris 2008.
%D A006037 R. K. Guy, Unsolved Problems in Number Theory, B2.
%D A006037 H. J. Hindin, Quasipractical numbers, IEEE Communications Magazine, March 
               1980, pp. 41-45.
%D A006037 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A006037 M. F. Hasler, <a href="b006037.txt">Table of n, a(n) for n=1,...,1000</
               a>.
%H A006037 Bj"orn B"ottcher, <a href="http://www-maths.swan.ac.uk/pgrads/bb/project/
               node37.html">Weird Numbers: Definition</a>
%H A006037 K. Uhland, <a href="http://uhlandkf.homestead.com/files/PuzzlePage/199611Pzl.htm">
               Weird Numbers</a>
%H A006037 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               WeirdNumber.html">Link to a section of The World of Mathematics.</
               a>
%H A006037 Wikipedia, <a href="http://en.wikipedia.org/wiki/Weird_number">Weird 
               number</a>
%t A006037 (* first do *) Needs["DiscreteMath`Combinatorica`"] (* then *) fQ[n_] 
               := Block[{d, l, t, i}, If[ DivisorSigma[1, n] > 2n && Mod[n, 6] != 
               0, d = Take[Divisors[n], {1, -2}]; l = 2^Length[d]; t = Table[ NthSubset[j, 
               d], {j, l - 1}]; i = 1; While[i < l && Plus @@ t[[i]] != n, i++ ]]; 
               If[i == l, True, False]]; Select[ Range[ 20000], fQ[ # ] &] (from 
               Robert G. Wilson v (rgwv(AT)rgwv.com), May 20 2005)
%o A006037 (PARI) isA006037(n,d=0)={ local(t); /* if d is not given, return nonzero 
               iff n is weird ; if d is given, return nonzero iff n is not the sum 
               of a subset of d */ if( !d, sigma(n)<=2*n & return /*must be abundant*/
               ; d=vecextract(divisors(n),"^-1")); setsearch( Set(d),n ) & return 
               /* equal to one element of d */; while( d[ #d]>n, d=vecextract(d,
               "^-1")); n >= (t = sum(i=1,#d, d[i])) & return( n-t /* nonzero if 
               n>t */ ); n > d[ #d] & ! isA006037( n - d[ #d], d=vecextract( d, 
               "^-1" )) & return; isA006037( n, d )}
%o A006037 (PARI) t=0; A006037=vector(1000,i,until( isA006037(t+=2),)= ; t) \\ - 
               M. F. Hasler, Mar 30 2008
%Y A006037 Cf. A002975, A005101, A005835, A005100, A138850; A087167.
%Y A006037 Sequence in context: A060541 A104475 A027804 this_sequence A002975 A061170 
               A125114
%Y A006037 Adjacent sequences: A006034 A006035 A006036 this_sequence A006038 A006039 
               A006040
%K A006037 nonn,nice
%O A006037 1,1
%A A006037 N. J. A. Sloane (njas(AT)research.att.com).
%E A006037 More terms from Jud McCranie (j.mccranie(AT)comcast.net), Oct 21 2001