A002975 Primitive weird numbers: weird numbers with no proper weird divisors.

70, 836, 4030, 5830, 7192, 7912, 9272, 10792, 17272, 45356, 73616, 83312, 91388, 113072, 243892, 254012, 338572, 343876, 388076, 519712, 539744, 555616, 682592, 786208, 1188256, 1229152, 1713592, 1901728, 2081824, 2189024, 3963968, 4128448

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

Sidney Kravitz notes that a(21) = 539744; it was misprinted as 539774 in the Benkoski & Erdős article. - Charles R Greathouse IV, Apr 04 2012

It appears that a weird number is primitive iff, divided by its largest prime factor, it is not weird. Is there a simple proof for this? - M. F. Hasler, Aug 20 2014 [The comment below does not answer this question.]

Yes, any primitive weird number, pwn, multiplied by any prime > sigma_1(pwn) is also weird. - Robert G. Wilson v, Jun 09 2015

A proper subsequence of A006037 and A091191. - Robert G. Wilson v, May 25 2015

Number of terms < 10^n: 0, 1, 2, 7, 13, 24, 48, 85, 152, 276, 499, 881, ..., . - Robert G. Wilson v, Jun 21 2017

The primitive weird number (pwn) 176405960704 is the least term which has as its abundance a pwn. Two other terms are 81152249741312, 14327148694372352. - Robert G. Wilson v, Sep 22 2017

Primitive weird numbers == 2 (mod 4): {70, 4030, 5830, 4199030, 1550860550, 66072609790, ...}. All the terms in A258374 appear so far. - Robert G. Wilson v, Nov 21 2015

See A258882 (and A258333) for terms of the form a(n)=2^k*p*q and A258401 for all other terms, with subsets A258883 (a(n)=2^k*p*q*r), A258884 (a(n)=2^k*p*q*r*s), A258885 (six distinct prime factors). A258374 and A258375 list the smallest terms with n prime factors (with / without counting multiplicity). - M. F. Hasler, Jul 12 2016

Sequence A273815 lists terms with nonsquarefree odd part, by definition excluded in A258883 and A258884. - M. F. Hasler, Feb 18 2018

Let n be a weird number and d be a divisor of n. If n/d is not weird, then either it is deficient or it is pseudoperfect. But if n/d is pseudoperfect, then multiplying the subset of the divisors of n/d that sums to n/d by d gives a solution for n, contradicting the assumption that n is weird. Therefore, n/d must be deficient. Of all the prime factors of n contributing to sigma(n)/n, the largest prime will contribute the least, and so if n/gpf(n) is deficient, then n/d is deficient for all divisors d of n, and n is a primitive weird number. - Charlie Neder, Oct 08 2018

The second part of the above reasoning is incorrect: gpf(n) may contribute more to sigma(n)/n than a smaller prime factor. For example, for n = 24, we have n/3 deficient, but n/2 abundant; for n = 350, n/7 is deficient but n/5 is abundant. - M. F. Hasler, Jan 25 2020

Examples

			10430 = A006037(8) is weird but not primitive weird because it has the proper weird divisor 70 = A006037(1).
		

References

• R. K. Guy, Unsolved Problems in Number Theory, B2.
• R. Honsberger, Mathematical Gems, M.A.A., 1973, p. 113.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..1161 (terms a(58)-a(160) from Donovan Johnson). [Term a(1159) inserted and b-file reformatted by _Georg Fischer_, Jan 16 2019]
• Stan Benkoski, Problem E2308, Amer. Math. Monthly, 79 (1972) 774.
• Gianluca Amato, Maximilian F. Hasler, Giuseppe Melfi and Maurizio Parton, Primitive weird numbers having more than three distinct prime factors, Riv. Mat. Univ. Parma, Vol. 7, No. 1 (2016) 153-163.
• Gianluca Amato, Maximilian F. Hasler, Giuseppe Melfi and Maurizio Parton, Primitive abundant and weird numbers with many prime factors, Journal of Number Theory vol. 201 (2019) pp. 436-459. DOI: 10.1016/j.jnt.2019.02.027. (Preprint: arXiv:1802.07178.)
• S. J. Benkoski and P. Erdős, On weird and pseudoperfect numbers, Math. Comp., 28 (1974), pp. 617-623. Alternate link; 1975 corrigendum.
• R. K. Guy, Letter to N. J. A. Sloane with attachment, Jun. 1991
• Douglas E. Iannucci, On primitive weird numbers of the form 2^k*p*q, arXiv:1504.02761 [math.NT], 2015.
• G. Melfi, On the conditional infiniteness of primitive weird numbers, Journal of Number Theory, Volume 147, February 2015, Pages 508-514.
• Robert G. Wilson v, Table of n, a(n) along with its abundance and their factorizations for n = 1..1161

Crossrefs

Cf. A006037.

Cf. A258882 and A258333; A258883, A258884, A258885 and A258401; A258374 and A258375.

Programs

Mathematica

(* first do *) << Combinatorica` (* then *) fQ[n_] := Block[{d = Most@ Divisors@ n, l = 2^(DivisorSigma[0, n] - 1), i = 1}, i = 1; While[i < l && Plus @@ NthSubset[i, d] != n, i++ ]; i == l]; lst = {}; Do[m = n; If[ Mod[n, 6] != 0 && DivisorSigma[1, n] > 2 n && Union[ Mod[ n, Join[lst, {n + 1}]]][[1]] != 0 && fQ@n, AppendTo[lst, n]; Print@n], {n, 2, 42000000, 2}] (* Robert G. Wilson v, Aug 04 2009 *)
(* Input: Range of even numbers --- Output: Primitive weird numbers *)
Block[{$RecursionLimit = Infinity},
  subOfSum[ss_, kk_, rr_] :=
   Module[{s = ss, k = kk, r = rr},
    If[s + w[[k]] >= mm && s + w[[k]] <= m, t = False;
     Goto[done] (* Found *),
     If[s + w[[k]] + w[[k + 1]] <= m,
      subOfSum[s + w[[k]], k + 1, r - w[[k]]]];
     If[s + r - w[[k]] >= m && s + w[[k + 1]] <= m,
      subOfSum[s, k + 1, r - w[[k]] ]]]; t]; (* end subOfSum *)
  greedyQ[ab_] := Module[{abn = ab, v, sum, s, j, jj, k}, tt = False;
    jj = Length[w]; (* start search *)
    Do[s = r; sum = 0; Do[v = w[[j]]; sum = sum + v;
      If[sum > abn, sum = sum - v; Goto[nxt]];
      If[sum == abn, tt = True; Goto[doneG]]; s = s - v;
      Label[nxt], {j, jj, 1, -1}];
     jj = jj - 1, {k, 1, jj - 1}]; Label[doneG];
    (* True means found, False not found *) tt]; (* end greedyQ *)
  cnt = 0;
  Do[ If[ Mod[n, 3] == 0, Goto[agn]]; r = DivisorSigma[1, n];
   m = r - 2*n;
   If[m > 0, fi = FactorInteger[n]; largestP = fi[[Length[fi]]][[1]];
    nn = n/largestP; If[m > 2*nn || Length[fi] < 3, Goto[agn]];
    If[DivisorSigma[1, nn] > 2*nn, Goto[agn]]; t = True; r = r - n;
    ww = Divisors[n]; lenW = Length[ww];
    Do[ If[ ww[[i]] <= m, w = Drop[ww, i - lenW]; Break[],
      r = r - ww[[i]]], {i, lenW - 1, 1, -1}];
    If[r >= m,
     If[ greedyQ[m], t = False, (* Powers of 2 dropped *)
      exp2 = fi[[1]][[2]]; sig2 = 2^(exp2 + 1) - 1; mm = m - sig2;
      lenW = Length[w]; ww = {};
      If[exp2 > 1,
       Do[ Do[ If[ w[[i]] == 2^ii, ww = AppendTo[ww, w[[i]]]],
      {i, 1, lenW}], {ii, 0, exp2}];
        w = Complement[w, ww]
       (* end T if *), w = Drop[w, 2]];
      (* end Pwr2 *) t = subOfSum[0, 1, r]]]; Label[done];
    If[t, Print[++cnt, "   ", n, "  ", t]]];
   Label[agn], {n, 2, 10000000, 2}]]
(* from Brent Baughn via http://mathematica.stackexchange.com/questions/73301/calculating-weird-numbers, Robert G. Wilson v, Nov 21 2015 *)

PARI

is_A002975(n)=is_A006037(n)&&!fordiv(n,d,!bittest(d,0)&&dA006037(d)&&return) \\ M. F. Hasler, Jan 07 2014

Extensions

More terms from Jud McCranie, Oct 21 2001

One more term from Robert G. Wilson v, Aug 04 2009

a(1)-a(123) double-checked by M. F. Hasler, Jan 07 2014

Edited by M. F. Hasler, Jul 12 2016