%I M5339 #166 Sep 08 2024 12:12:14

%S 70,836,4030,5830,7192,7912,9272,10430,10570,10792,10990,11410,11690,

%T 12110,12530,12670,13370,13510,13790,13930,14770,15610,15890,16030,

%U 16310,16730,16870,17272,17570,17990,18410,18830,18970,19390,19670

%N Weird numbers: abundant (A005101) but not pseudoperfect (A005835).

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

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

%C From _Alois P. Heinz_, Oct 30 2009: (Start)

%C 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:

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

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

%C There are no odd weird numbers < 1.8*10^19. - _Wenjie Fang_, Sep 04 2013

%C 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

%C No odd weird number exists below 10^21. This search was done on the volunteer computing project yoyo@home. - _Wenjie Fang_, Feb 23 2014

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

%C 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

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

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

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

%D David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 129.

%H Donovan Johnson, <a href="/A006037/b006037.txt">Table of n, a(n) for n = 1..10000</a> (first 4901 terms from Lukasz Swierczewski)

%H Gianluca Amato, Maximilian Hasler, Giuseppe Melfi, and Maurizio Parton, <a href="https://arxiv.org/abs/1803.00324">Primitive weird numbers having more than three distinct prime factors</a>, Riv. Mat. Univ. Parma, 7(1), (2016), 153-163, arXiv:1803.00324 [math.NT], 2018.

%H S. Benkoski, <a href="http://www.jstor.org/stable/2316276">Are All Weird Numbers Even?</a>, Problem E2308, Amer. Math. Monthly, 79 (7) (1972), 774.

%H S. J. Benkoski and P. Erdős, <a href="https://doi.org/10.1090/S0025-5718-1974-0347726-9">On weird and pseudoperfect numbers</a>, Math. Comp., 28 (1974), pp. 617-623. <a href="http://www.renyi.hu/~p_erdos/1974-24.pdf">Alternate link</a>; <a href="https://doi.org/10.1090/S0025-5718-1975-0360452-6">1975 corrigendum</a>.

%H David Eppstein, <a href="http://www.ics.uci.edu/~eppstein/numth/egypt/odd-one.html">Eqyptian Fractions</a>.

%H Wenjie Fang, <a href="https://arxiv.org/abs/2207.12906">Searching on the boundary of abundance for odd weird numbers</a>, arXiv:2207.12906 [math.NT], 2022.

%H R. K. Guy, <a href="/A001599/a001599_1.pdf">Letter to N. J. A. Sloane with attachment, Jun. 1991</a>.

%H H. J. Hindin, <a href="http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&amp;arnumber=1090205">Quasipractical numbers</a>, IEEE Communications Magazine, March 1980, pp. 41-45.

%H Odd Weird Search, <a href="https://www.rechenkraft.net/forum/viewtopic.php?f=57&amp;t=15230">Report on the recently completed batch</a>, Feb 23 2015.

%H OProject, <a href="http://web.archive.org/web/20131206135920/http://oproject.info/weird_list.php">Weird numbers list</a>.

%H J. Sandor and B. Crstici, <a href="http://bib.tiera.ru/ShiZ/math/other/Handbook%20Of%20Number%20Theory%20II%20-%20J.%20Sandor,%20B.%20Crstici%20(Kluwer,%202004).pdf">Handbook of number theory II</a>, chapter 1.8. [Broken link]

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WeirdNumber.html">Weird Number</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Weird_number">Weird number</a>.

%H Robert G. Wilson v, <a href="/A006037/a006037.pdf">Letter to N. J. A. Sloane, Jan. 1992</a>.

%H Robert G. Wilson v, <a href="/A007376/a007376.pdf">Letter to N. J. A. Sloane, Oct. 1993</a>.

%p isA006037 := proc(n)

%p isA005101(n) and not isA005835(n) ;

%p end proc:

%p for n from 1 do

%p if isA006037(n) then

%p print(n);

%p end if;

%p end do: # _R. J. Mathar_, Jun 18 2015

%t (* 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 *)

%o (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

%o (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

%o (PARI) t=0; A006037=vector(100,i, until( is_A006037(t+=2),); t) \\ _M. F. Hasler_, Mar 30 2008

%o (Haskell)

%o a006037 n = a006037_list !! (n-1)

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

%o -- _Reinhard Zumkeller_, Jan 21 2013

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

%Y Cf. A210455.

%K nonn,nice

%O 1,1

%A _N. J. A. Sloane_

%E More terms from _Jud McCranie_, Oct 21 2001