cp's OEIS Frontend

A006037 Weird numbers: abundant (A005101) but not pseudoperfect (A005835).

%I A006037 M5339 #167 Feb 16 2025 08:32:29
%S A006037 70,836,4030,5830,7192,7912,9272,10430,10570,10792,10990,11410,11690,
%T A006037 12110,12530,12670,13370,13510,13790,13930,14770,15610,15890,16030,
%U A006037 16310,16730,16870,17272,17570,17990,18410,18830,18970,19390,19670
%N A006037 Weird numbers: abundant (A005101) but not pseudoperfect (A005835).
%C A006037 OProject@Home in subproject Weird Engine calculates and stores the weird numbers.
%C A006037 There are no odd weird numbers < 10^17. - Robert A. Hearn (rah(AT)ai.mit.edu), May 25 2005
%C A006037 From _Alois P. Heinz_, Oct 30 2009: (Start)
%C A006037 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 A006037   1+5+7+10+14+35+298+10430 = 2+70+149+745+1043+1490+2086+5215
%C A006037   2+70+298+10430 = 1+5+7+10+14+35+149+745+1043+1490+2086+5215. (End)
%C A006037 There are no odd weird numbers < 1.8*10^19. - _Wenjie Fang_, Sep 04 2013
%C A006037 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 A006037 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 A006037 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 A006037 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 A006037 J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 70, p. 24, Ellipses, Paris 2008.
%D A006037 R. K. Guy, Unsolved Problems in Number Theory, B2.
%D A006037 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A006037 David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 129.
%H A006037 Donovan Johnson, <a href="/A006037/b006037.txt">Table of n, a(n) for n = 1..10000</a> (first 4901 terms from Lukasz Swierczewski)
%H A006037 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 A006037 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 A006037 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 A006037 David Eppstein, <a href="http://www.ics.uci.edu/~eppstein/numth/egypt/odd-one.html">Eqyptian Fractions</a>.
%H A006037 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 A006037 R. K. Guy, <a href="/A001599/a001599_1.pdf">Letter to N. J. A. Sloane with attachment, Jun. 1991</a>.
%H A006037 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 A006037 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 A006037 OProject, <a href="http://web.archive.org/web/20131206135920/http://oproject.info/weird_list.php">Weird numbers list</a>.
%H A006037 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 A006037 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WeirdNumber.html">Weird Number</a>.
%H A006037 Wikipedia, <a href="http://en.wikipedia.org/wiki/Weird_number">Weird number</a>.
%H A006037 Robert G. Wilson v, <a href="/A006037/a006037.pdf">Letter to N. J. A. Sloane, Jan. 1992</a>.
%H A006037 Robert G. Wilson v, <a href="/A007376/a007376.pdf">Letter to N. J. A. Sloane, Oct. 1993</a>.
%p A006037 isA006037 := proc(n)
%p A006037     isA005101(n) and not isA005835(n) ;
%p A006037 end proc:
%p A006037 for n from 1 do
%p A006037     if isA006037(n) then
%p A006037         print(n);
%p A006037     end if;
%p A006037 end do: # _R. J. Mathar_, Jun 18 2015
%t A006037 (* 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 A006037 (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 A006037 (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 A006037 (PARI) t=0; A006037=vector(100,i, until( is_A006037(t+=2),); t) \\ _M. F. Hasler_, Mar 30 2008
%o A006037 (Haskell)
%o A006037 a006037 n = a006037_list !! (n-1)
%o A006037 a006037_list = filter ((== 0) . a210455) a005101_list
%o A006037 -- _Reinhard Zumkeller_, Jan 21 2013
%Y A006037 Cf. A002975, A005101, A005835, A005100, A138850, A087167.
%Y A006037 Cf. A210455.
%K A006037 nonn,nice
%O A006037 1,1
%A A006037 _N. J. A. Sloane_
%E A006037 More terms from _Jud McCranie_, Oct 21 2001