A258333 -id:A258333 - cp's OEIS frontend

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

N. J. A. Sloane

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

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

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).

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

Cf. A006037.

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

(* 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 *)

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

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

A258882 Primitive weird numbers of the form 2^kpq with k > 0 and where p < q are odd primes.

Original entry on oeis.org

70, 836, 7192, 7912, 9272, 10792, 17272, 73616, 83312, 113072, 519712, 539744, 555616, 682592, 786208, 1188256, 1229152, 1901728, 2081824, 2189024, 3963968, 4128448, 4145216, 4486208, 4559552, 4632896, 4960448, 5440192, 5568448, 6460864, 6621632, 7354304, 7470272, 8000704, 8134208

Offset: 1

Douglas E. Iannucci and Robert G. Wilson v, Jun 14 2015

nonn

The number of terms < 10^n: 0, 1, 2, 5, 9, 15, 35, 61, 114, 204, 380, 696, 1703, 3548, 6726, 13137, ....
If 2^k*p*q is a weird number, it is necessarily primitive, and 2^(k+1) < p < 2^(k+2)-2 < q < 2^(2k+1).
No odd weird numbers are known and any even weird number must have at least 3 distinct prime factors, since all numbers of the form 2^k*p^m are deficient or pseudoperfect or perfect (iff m = 1 and p = 2^(k+1)-1 is a Mersenne prime). Sequence A258333 lists the number of terms in this sequence for given k. - M. F. Hasler, Jul 11 2016
Kravitz has shown that 2^k*p*q is a primitive weird number when the primes p and q satisfy p = (2^(k+1)*q-q-1)/(q+1-2^(k+1)). Many terms in this sequence are of this form, e.g., a(n) with n = 1, 2, 3, 4, 6, 7, 9, 10, 15, 23, 26, 38, 45, 75, 94, 144, 157, 187, 287, 327, 368, 370, 459, 607, 657, 658, .... Sequences A242025, A242998, ... are related to the special case where q is a Mersenne prime (A000668). - M. F. Hasler, Jul 12 2016
Weird numbers of the form 2^k*p*q are always primitive, so this condition could be omitted in the definition of this sequence. - M. F. Hasler, Jul 13 2016
About 35 years after Kravitz's work, the topic of weird numbers has regained interest after a CWU press release about students who used Kravitz's formula to find a large PWN of this form. See A242025 and A320875. - M. F. Hasler, Nov 20 2018

			a(1) = A002975(1) = 70 = 2*5*7.
a(2) = A002975(2) = 836 = 2^2*11*19.
A002975(3) = 4030 = 2*5*13*31 is not in this sequence since it is not of the required form.
The same is true for A002975(4) = 5830.
a(3) = A002975(5) = 7192 = 2^3*29*31, etc.
A002975(179) = 2319548096 = 2^6 * 137^2 * 1931 is the first term of A002975 with only two odd prime divisors, but not of the required form. - _M. F. Hasler_, Nov 20 2018

S. Kravitz, A search for large weird numbers. J. Recreational Math. 9 (1976), 82-85 (1977). Zbl 0365.10003

Douglas E. Iannucci and Robert G. Wilson v, Table of n, a(n) for n = 1..15384, updated Dec 06 2015; corrected by _M. F. Hasler_, Jul 16 2016
R. Bagula et al., A very big weird number, Number Theory group on LinkedIn (web.archive.org snapshot; page no longer available). Dec. 2013
Central Washington University, CWU Math Students Break World Record for Largest Weird Number [alternate article]
Douglas E. Iannucci, On primitive weird numbers of the form 2^k*p*q, arXiv:1504.02761 [math.NT], 2015.
Giuseppe Melfi, On the conditional infiniteness of primitive weird numbers, Journal of Number Theory, Vol. 147, Feb 2015, pp 508-514.
Eric Weisstein's World of Mathematics, Weird Number.
Wikipedia, Weird number

Cf. A002975, A258401 (PWN not of this form), A258374, A258375, A258883, A258884, A258885.
Cf. A242025, A242993, A242998, A242999, A243003 (related to the subsequence with q = (2^k*p-p-1)/(p+1-2^k) and p a Mersenne prime in A000668).
Cf. A320875 (more general case of Karavitz' formula).

(* copy the terms from A002975, assign them equal to 'lst' and then *) fQ[n_] := Block[{m = n}, While[ Mod[m, 2] == 0, m /= 2]; PrimeOmega@ m == 2]; Select[lst, fQ]

select(t->factor(t)[, 2][^1]=[1, 1]~, A002975) \\ Assuming that A002975 is defined as set or vector. - M. F. Hasler, Jul 11 2016

A258882 union A258401 is A002975.

Edited by M. F. Hasler, Jul 11 2016, Nov 20 2018

A275491 Number of primitive weird numbers (A002975) below 10^n.

Original entry on oeis.org

0, 0, 1, 2, 7, 13, 24, 48, 85, 152, 276, 499, 881

Offset: 0

M. F. Hasler, Jul 30 2016

It is not known unconditionally whether there are infinitely many primitive weird numbers (PWN; A002975), although numerical data provides strong evidence: even the number of weird numbers of the form 2^k*p*q (A258882, A258333) seems to increase rapidly as k increases. G. Melfi has shown that Cramer's conjecture implies the infiniteness of PWN.

Partial sums of A275492.

Giuseppe Melfi, On the conditional infiniteness of primitive weird numbers, Journal of Number Theory, Volume 147, February 2015, Pages 508-514.

Cf. A002975, A006037, A258333, A258882, A275492, A275493, A275494.

my(s=0); vector(10,n,s+=sum(n=10^n\20+1,10^n\2,is_A002975(n*2)))

vector(10,n,#select(t->t<10^n,A002975)) \\ If A002975 is defined as set, vector, or list with enough terms.

a(12) from Robert G. Wilson v, May 25 2018

a(10) corrected by Amiram Eldar, Sep 02 2023

A320875 Least d > 0 such that both Q = M + 2d and R = M + (M^2-1)/(Q-M) are prime, where M = 2^n - 1 = A000225(n), or 0 if there is no such d.

Original entry on oeis.org

0, 1, 2, 1, 3, 8, 2, 0, 6, 4, 66, 16, 20, 0, 6, 1, 2720, 0, 32, 0, 164, 8, 0, 524288, 153, 3573184, 2097152, 7354396, 19436, 4517888, 672, 0, 174080, 0, 262146, 1984, 48, 0, 4194296, 79, 30720, 128, 1825866, 4188889, 194396, 27227248, 0, 16384, 723, 0, 265227072, 22771712, 13982720, 134217728, 59885796, 587144, 19436, 0, 17179869152, 8388608

Offset: 1

M. F. Hasler, Nov 11 2018

nonn

It is easy to see that R can't be an integer unless M < Q < M^2 + M.
Nonzero terms yield primitive weird numbers (PWN) 2^(n-1)*Q*R, cf. A258882.
This idea was used by S. Kravitz in 1976 and 35 years later by students of CWU to find the largest known PWN, cf. links and A242025, A242993, A242998, A242999, A243003. The 226 digits mentioned in the news article correspond not to a PWN but to the prime R for a(381) = 5456. The corresponding prime Q = M(381) + 2*5456 is the 54th prime after M(381), and only the third one for which R is an integer. The 127 digit PWN they found earlier corresponds to a non-minimal solution d = 34008 for n = 109. (It is a matter of seconds to find many much larger solutions, see examples.) This news led to renewed interest in this topic and a series of recent research papers, see references in A258882 and A002975.
Sequences A242025, A242993, A242998, A242999, A243003 consider PWN of the form 2^(k-1)*Q*R(k,Q) where the prime Q is fixed to be a Mersenne prime A000668, and k is varied to find a prime R.
Zero terms do not mean that there aren't PWN of the form 2^(n-1)*p*q with M+1 = 2^n < p < 2M < q < M(M+1). For example, a(8) = 0, but there are A258333(8) = 53 weird numbers with such (p,q). However, the two primes never satisfy the relation (p-M)(q-M) = M^2-1 which is considered here for (Q,R). - M. F. Hasler, Nov 20 2018

			a(109) = 8436 yields a 62-digit prime R and a 127 digit PWN 2^108*Q*R.
a(381) = 5456 yields a 226-digit prime R and a 455 digit PWN 2^380*Q*R. (This and the preceding one are mentioned in the News articles, cf LINKS.)
a(391) = 16386 leads to a 231-digit prime R and a 466-digit PWN 2^390*Q*R.
a(409) = 12360 leads to a 242-digit prime R and a 488-digit PWN 2^408*Q*R.
a(421) = 1661 leads to a 250-digit prime R and a 504-digit PWN 2^420*Q*R.
a(430) = 10304 leads to a 255-digit prime R and a 514-digit PWN 2^429*Q*R.
a(441) = 36080 leads to a 261-digit prime R and a 526-digit PWN 2^440*Q*R.
a(505) = 20726 leads to a 300-digit prime R and a 604-digit PWN 2^504*Q*R.

Daily Record, CWU students find longest 'weird' number, and also Yakima Herald, CWU math students calculate what no mathematician has before (backup on web.archive.org, page no longer available), both from Dec. 5, 2013
S. Kravitz, A search for large weird numbers. J. Recreational Math. 9(1976), 82-85 (1977). Zbl 0365.10003

Cf. A258882, subsequence of A002975.

Cf. A242025, A242993, A242998, A242999, A243003 (all related to the case Q = 2^p-1 in A000668, p in A000043).

a(n)={my(M=2^n-1,S=M^2-1); fordiv(S+!S,D, ispseudoprime(M+D)&& ispseudoprime(M+S/D)&& return(D/2))} \\ Much faster than the variant below, but requires increasingly more stack space (allocatemem()) for larger n.

A320875(n,L=0)={my(M=2^n-1,S=M^2-1); forprime(Q=M+1,if(L,L,M<

A265727 Least primitive weird number, pwn, (A002975) which is divisible by the n-th prime (A000040).

Original entry on oeis.org

70, 70, 836, 4030, 17272, 836, 7912, 7192, 4030, 113072, 83312, 7912, 8812312, 5830, 4199030, 9272, 91388, 10792, 23941578736, 786208, 682592, 569494624, 555616, 539744, 15126992, 73616, 519712

Offset: 3

Douglas E. Iannucci and Robert G. Wilson v, Dec 14 2015

nonn

No odd weird number exists below 10^21. The search is done on the volunteer computing project yoyo@home. - Wenjie Fang, Feb 23 2014

As of Dec 14 2015, there is no known pwn which is divisible by 3. Therefore the offset denotes the third prime number, 5.

			a(6) is 4030 since it is the first pwn to be divisible by the sixth prime number, 13. 4030 = 13 * 310.

Douglas E. Iannucci, On primitive weird numbers of the form 2^k*p*q, arXiv:1504.02761 [math.NT], 2015.
Linked In, Number Theory, A very big weird number
Giuseppe Melfi, On the conditional infiniteness of primitive weird numbers, Journal of Number Theory, Vol. 147, Feb 2015, pp. 508-514.
Wikipedia, Weird number.

Cf. A002975, A258250, A258333, A258374, A258375, A258401, A258882, A258883, A258884, A258885, A265726, A265728.

(* copy the terms from A002975, assign them equal to 'lst' and then *) f[n_] := Select[lst, Mod[#, Prime@ n] == 0 &][[1]]; Array[f, 27, 3]

A275492 Number of primitive weird numbers (A002975) between 10^n and 10^(n+1).

Original entry on oeis.org

0, 1, 1, 5, 6, 11, 24, 37, 67, 124, 223, 382

Offset: 0

M. F. Hasler, Jul 30 2016

It is not known unconditionally whether there are infinitely many primitive weird numbers (PWN, A002975), although numerical data provides strong evidence: even the number of weird numbers of the form 2^k*p*q (A258882, A258333) seems to increase rapidly as k increases. Melfi has shown that Cramer's conjecture implies the infiniteness of PWN.

Giuseppe Melfi, On the conditional infiniteness of primitive weird numbers, Journal of Number Theory, Volume 147, February 2015, Pages 508-514.

Cf. A002975, A006037, A258333, A258882, A275491, A275493, A275494.

a(n)=sum(n=10^n\2+1,5*10^n,is_A002975(n*2))

Wrong initial term removed and a(11) added by Amiram Eldar, Sep 02 2023

A275493 Number of primitive weird numbers (A002975) below 2^n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 3, 6, 8, 9, 10, 14, 16, 20, 24, 29, 33, 46, 55, 61, 79, 87, 103, 128, 152, 173, 234, 266, 313, 403, 483, 561, 756, 850, 940

Offset: 0

M. F. Hasler, Jul 30 2016

Giuseppe Melfi, On the conditional infiniteness of primitive weird numbers, Journal of Number Theory, Volume 147, February 2015, Pages 508-514.

Cf. A002975, A006037, A258333, A258882, A275494, A275491, A275492.

vector(40,n,#select(t->t<2^n,A002975)) \\ assuming that A002975 is defined as a set, vector or list with enough terms.

a(0) inserted, a(39) corrected, and a(40) added by Amiram Eldar, Sep 02 2023

A275494 Number of primitive weird numbers (A002975) between 2^n and 2^(n+1).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 3, 2, 1, 1, 4, 2, 4, 4, 5, 4, 13, 9, 6, 18, 8, 16, 25, 24, 21, 61, 32, 47, 90, 80, 78, 195, 94, 90

Offset: 0

M. F. Hasler, Jul 30 2016

			The first primitive weird numbers are 70, 836, 4030, 5830, 7192, 7912, 9272, 10792, ..., so there is one between 2^6 and 2^7 = 128, one between 2^9 and 2^10 = 1024, one between 2^11 and 2^12 = 4096, three between 2^12 and 2^13, etc.

Giuseppe Melfi, On the conditional infiniteness of primitive weird numbers, Journal of Number Theory, Volume 147, February 2015, Pages 508-514.

Cf. A002975, A006037, A258333, A258882, A275493, A275491, A275492.

PARI
```
a(n)=sum(n=2^n\2+1,2^n,is_A002975(n*2))
```

a(n) = A275493(n+1) - A275493(n).

a(39) from Amiram Eldar, Sep 02 2023

A265726 Primitive weird numbers whose abundance is a record.

Original entry on oeis.org

70, 836, 7192, 9272, 73616, 243892, 338572, 1188256, 1901728, 3963968, 28279232, 36228736, 91322752, 141659096, 263144192, 351295232, 664373504, 2113834496, 5522263024, 6933503488, 19179527168, 22755515392, 31574500724, 98620009472, 135895635968

Offset: 1

Douglas E. Iannucci and Robert G. Wilson v, Dec 14 2015

nonn

Although the abundance A(n) = sigma(n) - 2n is increasing, the (relative) abundancy sigma(n)/n is decreasing, except at indices {3, 6, 8, 15, 16, 19, 24 ...}. No term has larger abundancy than 2 + 2/35, that of a(1). - M. F. Hasler, Nov 14 2018

			a(1) = 70 since it is the first term in A002975; its abundance is 4.
a(2) is 836 since its abundance, 8, exceeds that of a(1); 4.
a(3) is 7192 = A002975(5) since its abundance, 16, exceeds that of a(2) and that of A002975(1..4).

Douglas E. Iannucci, On primitive weird numbers of the form 2^k*p*q, arXiv:1504.02761 [math.NT], 2015.
Giuseppe Melfi, On the conditional infiniteness of primitive weird numbers, Journal of Number Theory, Vol. 147, Feb 2015, pgs 508-514.
Wikipedia, Weird number

Cf. A002975, A258250, A258333, A258374, A258375, A258401, A258882, A258883, A258884, A258885, A265727, A265728.

(* copy the terms from A002975, assign them equal to 'lst' and then *) f[n_] := DivisorSigma[1, n] - 2n; k = 1; lsu = {}; mx = 0; While[k < 647, ds = f@ lst[[k]]; If[ds > mx, mx = ds; AppendTo[lsu, lst[[k]]]]; k++]; lsu

A265728 Least primitive weird number, pwn, (A002975) whose abundance is divisible by the n-th prime (A000040), or 0 if no such pwn exists.

Original entry on oeis.org

70, 232374697216, 73616, 9272, 243892, 343876, 4128448, 519712, 1901728, 338572, 5568448, 6621632, 272240768, 4960448, 7470272, 1673087984, 146279296, 5440192, 91322752, 8134208, 35442304, 286717696, 54962343424, 110232704, 6460864, 2812606976, 44473216, 141659096, 33736064, 58668928, 9537494528, 37499776, 292335872, 795730688, 530110208, 18657360896, 16995175424, 664373504, 266311424, 23049995264, 15152370176, 17124699136, 64015565312, 52059008

Offset: 1

Douglas E. Iannucci and Robert G. Wilson v, Dec 14 2015

nonn

No odd weird number exists below 10^21. The search is done on the volunteer computing project yoyo@home. - Wenjie Fang, Feb 23 2014

			a(1) = 70 since it is the least pwn whose abundance, 4, is divisible by the first prime, 2.
a(2) = 0 since there is no known odd pwn and if there were, there is no reason why the abundance would be == 0 (mod 3).
a(3) = 73616 since it is the first pwn whose abundance, 80, is divisible by the third prime, 5.

Douglas E. Iannucci, On primitive weird numbers of the form 2^k*p*q, arXiv:1504.02761 [math.NT], 2015.
Linked In, Number Theory, A very big weird number
Giuseppe Melfi, On the conditional infiniteness of primitive weird numbers, Journal of Number Theory, Vol. 147, Feb 2015, pp. 508-514.
Wikipedia, Weird number

Cf. A002975, A258250, A258333, A258374, A258375, A258401, A258882, A258883, A258884, A258885, A265726, A265727.

(* copy the terms from A002975, assign them equal to 'lst' and then *) f[n_] := Select[lst, Mod[ DivisorSigma[1, #] - 2#, Prime@ n] == 0 &][[1]]; Array[f, 30]

cp's OEIS Frontend

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

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A258882 Primitive weird numbers of the form 2^k*p*q with k > 0 and where p < q are odd primes.

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A275491 Number of primitive weird numbers (A002975) below 10^n.

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A320875 Least d > 0 such that both Q = M + 2d and R = M + (M^2-1)/(Q-M) are prime, where M = 2^n - 1 = A000225(n), or 0 if there is no such d.

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A265727 Least primitive weird number, pwn, (A002975) which is divisible by the n-th prime (A000040).

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A275492 Number of primitive weird numbers (A002975) between 10^n and 10^(n+1).

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A275493 Number of primitive weird numbers (A002975) below 2^n.

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A258882 Primitive weird numbers of the form 2^kpq with k > 0 and where p < q are odd primes.