1,1

There are no odd weird numbers < 10^17. - Robert A. Hearn (rah(AT)ai.mit.edu), May 25 2005

Contribution from Alois P. Heinz (heinz(AT)hs-heilbronn.de), Oct 30 2009: (Start)

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:

1+5+7+10+14+35+298+10430 = 2+70+149+745+1043+1490+2086+5215

2+70+298+10430 = 1+5+7+10+14+35+149+745+1043+1490+2086+5215 (End)

S. Benkoski, "Are All Weird Numbers Even?", Problem E2308, Amer. Math. Monthly, 79 (1972), 774.

S. J. Benkoski and P. Erdos, On weird and pseudoperfect numbers, Math. Comp., 28 (1974), 617-623.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 70, p. 24, Ellipses, Paris 2008.

R. K. Guy, Unsolved Problems in Number Theory, B2.

H. J. Hindin, Quasipractical numbers, IEEE Communications Magazine, March 1980, pp. 41-45.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

M. F. Hasler, Table of n, a(n) for n=1,...,1000.

Bj"orn B"ottcher, Weird Numbers: Definition

K. Uhland, Weird Numbers

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Wikipedia, Weird number

(* 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)

(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 )}

(PARI) t=0; A006037=vector(1000, i, until( isA006037(t+=2), )= ; t) \\ - M. F. Hasler, Mar 30 2008

Cf. A002975, A005101, A005835, A005100, A138850; A087167.

Sequence in context: A060541 A104475 A027804 this_sequence A002975 A061170 A125114

Adjacent sequences: A006034 A006035 A006036 this_sequence A006038 A006039 A006040

nonn,nice,new

N. J. A. Sloane (njas(AT)research.att.com).

More terms from Jud McCranie (j.mccranie(AT)comcast.net), Oct 21 2001