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A006037
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Weird numbers: abundant (A005101) but not pseudoperfect (A005835).
(Formerly M5339)
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17
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70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, 10990, 11410, 11690, 12110, 12530, 12670, 13370, 13510, 13790, 13930, 14770, 15610, 15890, 16030, 16310, 16730, 16870, 17272, 17570, 17990, 18410, 18830, 18970, 19390, 19670
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| There are no odd weird numbers < 10^17. - Robert A. Hearn (rah(AT)ai.mit.edu), May 25 2005
Contribution from Alois P. Heinz, 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)
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REFERENCES
| 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).
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LINKS
| M. F. Hasler, Table of n, a(n) for n=1,...,1000.
S. Benkoski, Are All Weird Numbers Even?, Problem E2308, Amer. Math. Monthly, 79 (7) (1972), 774.
S. J. Benkoski and P. Erdos, On weird and pseudoperfect numbers, Math. Comp., 28 (1974), 617-623
David Eppstein, Eqyptian Fractions
J. Sandor and B Crstici, Handbook of number theory II, chapter 1.8.
Eric Weisstein's World of Mathematics, Weird Number
Wikipedia, Weird number
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MATHEMATICA
| (* 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, May 20 2005 *)
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PROG
| (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
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CROSSREFS
| Cf. A002975, A005101, A005835, A005100, A138850; A087167.
Sequence in context: A199829 A027804 A177298 * A002975 A061170 A125114
Adjacent sequences: A006034 A006035 A006036 * A006038 A006039 A006040
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KEYWORD
| nonn,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Jud McCranie (JudMcCranie(AT)ugaalum.uga.edu), Oct 21 2001
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