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A006037 Weird numbers: abundant (A005101) but not pseudoperfect (A005835).
(Formerly M5339)
24
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; text; internal format)
OFFSET

1,1

COMMENTS

OProject@Home in subproject Weird Engine calculates and stores the weird numbers.

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

From Alois P. Heinz, Oct 30 2009: (Start)

The first weird number that has more than one decomposition of its 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)

There are no odd weird numbers < 1.8*10^19. - Wenjie Fang, Sep 04 2013

S. Benkowski and P. Erdős (1974) proved that the asymptotic density W of weird numbers is positive. It can be shown that W < 0.0101 (see A005835). - Jaycob Coleman, Oct 26 2013

No odd weird number exists below 10^21. The search is done on the volunteer computing project yoyo@home. - Wenjie Fang, Feb 23 2014

No odd weird number with abundance less than 10^14 exists below 10^28. See Odd Weird Search link - Wenjie Fang, Feb 25 2015

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.

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

LINKS

L. Swierczewski and Donovan Johnson, Table of n, a(n) for n = 1..10000 (first 4901 terms from L. Swierczewski)

S. Benkoski, Are All Weird Numbers Even?, Problem E2308, Amer. Math. Monthly, 79 (7) (1972), 774

S. J. Benkoski and P. Erdős, On weird and pseudoperfect numbers, Math. Comp., 28 (1974), pp. 617-623. Alternate link; 1975 corrigendum

David Eppstein, Eqyptian Fractions

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

Odd Weird Search, Report on the recently completed batch, Feb 23 2015.

OProject, Weird numbers list

J. Sandor and B. Crstici, Handbook of number theory II, chapter 1.8.

Eric Weisstein's World of Mathematics, Weird Number

Wikipedia, Weird number

MAPLE

isA006037 := proc(n)

    isA005101(n) and not isA005835(n) ;

end proc:

for n from 1 do

    if isA006037(n) then

        print(n);

    end if;

end do: # R. J. Mathar, Jun 18 2015

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[ # ] &] (* Robert G. Wilson v, May 20 2005 *)

PROG

(PARI) is_A006037(n, d, s)={s || (n<s=sum(i=1, #d=divisors(n)[^-1], d[i])) || return; !setsearch(d, n) && !while(s>n && d[#d]>n, s-=d[#d]; d=d[^-1]) && if(s>n, is_A006037(n - d[#d], d[^-1], s-d[#d]) && is_A006037(n, d[^-1], s-d[#d]), s<n)} \\ Returns true iff n is weird, or, iff d and s=sum(d_i) are given, if n is not the sum if a subset of d. Improved using more recent PARI syntax by M. F. Hasler, Jul 15 2016

(PARI) is_A006037(n, d=divisors(n)[^-1], s=sum(i=1, #d, d[i]))={s>n && !is_A005835(n, d, s)} \\ It is shorter, avoids content duplication, and also computationally slightly more efficient to implement this using is_A005835() with the same optional arguments. Therefore we prefer this non self contained version. - M. F. Hasler, Jul 17 2016

(PARI) t=0; A006037=vector(100, i, until( is_A006037(t+=2), ); t) \\ M. F. Hasler, Mar 30 2008

(Haskell)

a006037 n = a006037_list !! (n-1)

a006037_list = filter ((== 0) . a210455) a005101_list

-- Reinhard Zumkeller, Jan 21 2013

CROSSREFS

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

Cf. A210455.

Sequence in context: A212236 A177298 A230897 * A002975 A258250 A258882

Adjacent sequences:  A006034 A006035 A006036 * A006038 A006039 A006040

KEYWORD

nonn,nice,changed

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Jud McCranie, Oct 21 2001

STATUS

approved

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Last modified July 30 08:31 EDT 2016. Contains 275173 sequences.