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A006037
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Weird numbers: abundant (A005101) but not pseudoperfect (A005835).
(Formerly M5339)
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70
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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;
text;
internal format)
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OFFSET
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1,1
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COMMENTS
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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
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. This search was 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
A weird number k multiplied by a prime p > sigma(k) is again weird. Primitive weird numbers (A002975) are those which are not a multiple of a smaller term, i.e., don't have a weird proper divisor. Sequence A065235 lists odd numbers that can be written in only one way as sum of their divisors, and A122036 lists those which are not in A136446, i.e., not sum of proper divisors > 1. - M. F. Hasler, Jul 30 2016
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REFERENCES
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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).
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LINKS
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MAPLE
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isA006037 := proc(n)
isA005101(n) and not isA005835(n) ;
end proc:
for n from 1 do
if isA006037(n) then
print(n);
end if;
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MATHEMATICA
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(* 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 *)
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PROG
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(PARI) is_A006037(n, d=divisors(n), s=vecsum(d)-n, m=#d-1)={ m||return; while(d[m]>n, s-=d[m]; m--); d[m]<n && if(s>n, is_A006037(n-d[m], d, s-d[m], m-1) && is_A006037(n, d, s-d[m], m-1), s<n && m<#d-1)} \\ M. F. Hasler, Mar 30 2008; improved and updated to current PARI syntax by M. F. Hasler, Jul 15 2016
(PARI) is_A006037(n, d=divisors(n)[^-1], s=vecsum(d))={s>n && !is_A005835(n, d, s)} \\ Equivalent but slightly faster than the self-contained version above.-- For efficiency, ensure that the argument is even or add "!bittest(n, 0) && ..." to check this first. - M. F. Hasler, Jul 17 2016
(Haskell)
a006037 n = a006037_list !! (n-1)
a006037_list = filter ((== 0) . a210455) a005101_list
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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