The OEIS is supported by the many generous donors to the OEIS Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A006037 Weird numbers: abundant (A005101) but not pseudoperfect (A005835). (Formerly M5339) 62
 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. 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 n multiplied by a prime p > sigma(n) 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 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) Gianluca Amato, Maximilian Hasler, Giuseppe Melfi and Maurizio Parton, Primitive weird numbers having more than three distinct prime factors, Riv. Mat. Univ. Parma, 7(1), (2016), 153-163, arXiv:1803.00324 [math.NT], 2018. 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 Wenjie Fang, Searching on the boundary of abundance for odd weird numbers, arXiv:2207.12906 [math.NT], 2022. 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 Robert G. Wilson v, Letter to N. J. A. Sloane, Jan. 1992 Robert G. Wilson v, Letter to N. J. A. Sloane, Oct. 1993 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=divisors(n), s=vecsum(d)-n, m=#d-1)={ m||return; while(d[m]>n, s-=d[m]; m--); d[m]n, is_A006037(n-d[m], d, s-d[m], m-1) && is_A006037(n, d, s-d[m], m-1), sn && !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 (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 A354282 A329190 Adjacent sequences:  A006034 A006035 A006036 * A006038 A006039 A006040 KEYWORD nonn,nice AUTHOR EXTENSIONS More terms from Jud McCranie, Oct 21 2001 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified October 2 15:42 EDT 2022. Contains 357226 sequences. (Running on oeis4.)