A258882 -id:A258882 - 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

A125247 Numbers n whose abundance sigma(n) - 2n = -8. Numbers n whose deficiency is 8.

Original entry on oeis.org

22, 130, 184, 1012, 2272, 18904, 33664, 70564, 85936, 100804, 391612, 527872, 1090912, 17619844, 2147713024, 6800695312, 34360655872, 549759483904, 1661355408388, 28502765343364, 82994670582016, 99249696661504, 120646991405056, 431202442356004, 952413274955776

Offset: 1

Jason G. Wurtzel, Nov 25 2006

a(19) > 10^12. - Donovan Johnson, Dec 08 2011
a(20) > 10^13. - Giovanni Resta, Mar 29 2013
a(30) > 10^18. - Hiroaki Yamanouchi, Aug 21 2018
a(20) <= 36028797958488064 ~ 3.6*10^16. Indeed, if k is in A057195 then 2^(k-1)*A168415(k) is in this sequence, and k=28 yields this upper bound for a(20) which is in any case a term of this sequence. - M. F. Hasler, Apr 27 2015
If n is in this sequence and p a prime not dividing n, then np is abundant if and only if p < sigma(n)/8 = n/4-1. For all n=a(k) except {22, 70564, 100804, 17619844}, there is such a p near this limit, such that n*p is a primitive weird number (A002975; in A258882 for the terms mentioned in the preceding comment). - M. F. Hasler, Jul 20 2016
Any term x of this sequence can be combined with any term y of A088833 to satisfy the property (sigma(x)+sigma(y))/(x+y) = 2, which is a necessary (but not sufficient) condition for two numbers to be amicable. - Timothy L. Tiffin, Sep 13 2016
Is there any odd number in this sequence? Is it possible to prove the contrary? - M. F. Hasler, Feb 22 2017

			The abundance of 22 = (1+2+11+22)-44 = -8

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..29

Cf. A033880, A088833 (abundance 8).

[n: n in [1..2*10^7] | (DivisorSigma(1,n)-2*n) eq - 8]; // Vincenzo Librandi, Jul 22 2016

Select[Range[10^6], DivisorSigma[1, #] - 2 # == -8 &] (* Michael De Vlieger, Jul 21 2016 *)

for(n=1,1000000,if(((sigma(n)-2*n)==-8),print1(n,",")))

a(13)-a(15) from Klaus Brockhaus, Nov 29 2006
a(16)-a(17) from Donovan Johnson, Dec 23 2008
a(18) from Donovan Johnson, Dec 08 2011
a(19) from Giovanni Resta, Mar 29 2013
a(20)-a(25) from Hiroaki Yamanouchi, Aug 21 2018

A258883 Primitive weird numbers (PWN) of the form 2^kpq*r with k > 0 and where p < q < r are odd primes.

Original entry on oeis.org

4030, 5830, 45356, 91388, 243892, 254012, 338572, 343876, 388076, 1713592, 8812312, 9928792, 11339816, 11547352, 15126992, 17999992, 29581424, 38546576, 74899952, 85389368, 89283592, 95327216, 141659096, 146764264, 162079768, 173482552, 569494624, 632874016

Offset: 1

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

nonn

The condition k > 0 is not really a limitation since a product of three odd primes cannot be weird. -- Numbers of the form 2^k*p^2*q having only two distinct odd prime divisors, e.g., A258401(45) = 2319548096 = 2^6 * 137^2 * 1931 or A258401(143) = 232374697216 = 2^8 * 797^2 * 1429, are neither in A258882 nor in the present sequence as it is currently defined, although they are in the set of weird numbers 2^k*p*q*r with odd primes p,q,r. (PWN with nonsquarefree odd part are listed in A273815.) - M. F. Hasler, Jul 18 2016, amended Nov 09 2017

It appears that there are (2, 7, 12, 18, 41, ...) terms with k = valuation(a(n),2) = 1, 2, 3, etc. The smallest and largest such are (4030, 45356, 1713592, 15126992, 569494624, 5353519168, 96743686016, 1009572479744, ...) resp. (5830, 388076, 173482552, 6587973136, 297512429728, ...). - M. F. Hasler, Nov 09 2017

			a(1) = 4030 = 2*5*13*31.
a(2) = 5830 = 2*5*11*53.
a(3) = 45356 = 2^2*17*23*29.

M. F. Hasler, Table of n, a(n) for n = 1..121 (Using A002975(1..1073) calculated by _Robert G. Wilson v_.)

Cf. A002975, A258401, A258882, A258884, A258885.

(* copy the terms from A002975, assign them to 'lst' and then *) Select[ lst, PrimeNu@# == 4 &] (* WARNING: this code selects PWN with 3 distinct odd prime factors but does not exclude that they occur with multiplicity > 1, which is forbidden by definition of this sequence. - M. F. Hasler, Jul 12 2016 *)

select(w->factor(w)[,2][^1]~==[1,1,1], A002975) \\ Assuming that A002975 is defined as set or vector. - M. F. Hasler, Jul 12 2016

A258401 Primitive weird numbers (A002975) of the form 2^kpq*x with k >= 0 and odd p, q, x >= 3.

Original entry on oeis.org

4030, 5830, 45356, 91388, 243892, 254012, 338572, 343876, 388076, 1713592, 4199030, 8812312, 9928792, 11339816, 11547352, 15126992, 17999992, 29465852, 29581424, 38546576, 74899952, 85389368, 89283592, 95327216, 120888092, 141659096, 146764264, 162079768, 173482552

Offset: 1

Robert G. Wilson v, May 28 2015

nonn

The complement of A258882 in A002975, i.e., primitive weird numbers not of the form 2^k*p*q with primes p, q. Equivalently, subsequence of A002975 for numbers with at least 3 odd prime factors, counting multiplicity. (No weird number is of the form 2^k*p^m.) Note that, e.g., a(40) = 2^6 * 137^2 * 1931 and a(143) = 2^8 * 797^2 * 1429 have only 3 distinct prime factors.
Primitive weird numbers of the excluded set (of the form 2^k*p*q, cf. A258882) are well studied and comparably easier to produce, see the Douglas E. Iannucci link; therefore this sequence is noteworthy and harder to produce.
More rare are the primitive weird numbers in which there is an odd prime squared factor, for example:
a(40) = A002975(156) = 1550860550 = 2 * 5^2 * 29 * 37 * 137 * 211,
a(45) = A002975(179) = 2319548096 = 2^6 * 137^2 * 1931,
a(117) = A002975(483) = 66072609790 = 2 * 5 * 11 * 127^2 * 167 * 223,
a(123) = A002975(508) = 114141404156 = 2^2 * 13^2 * 19 * 383 * 23203,
a(143) = A002975(725) = 232374697216 = 2^8 * 797^2 * 1429.
These PWN with an odd square factor are now listed as A273815. - M. F. Hasler, Jul 10 2016

Robert G. Wilson v, Table of n, a(n) for n = 1..186 (52 terms were from M. F. Hasler)
Douglas E. Iannucci, On primitive weird numbers of the form 2^k*p*q, arXiv:1504.02761 [math.NT], 2015.

Cf. A002975, A258401, A258374, A258375, A258882, A258883, A258884, A258885.

(* copy the terms from A002975, assign them equal to 'pwn' and then *) fQ[n_] :=
Block[{m = n}, While[ Mod[m, 2] == 0, m /= 2]; Total[Last@# & /@ FactorInteger@ m] > 2]; Select[pwn, fQ] (* Robert G. Wilson v, May 28 2015 and modified Mar 30 2017 *)

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

Edited and definition corrected by M. F. Hasler, Jul 10 2016

A258884 Primitive weird numbers, pwn, of the form 2^kpqrs with k > 0 and where p < q < r < s are odd primes.

Original entry on oeis.org

4199030, 29465852, 120888092, 259858324, 260378492, 410832532, 775397948, 785187524, 903217276, 989226964, 1609445332, 2358115084, 3254323124, 3381352084, 3381872252, 3781448788, 3782267372, 5056717796, 5065605532, 5066413508, 9210347984, 9772585048, 9776437996, 9843119884

Offset: 1

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

nonn

This differs from the sequence of primitive weird numbers with 5 (or 4 odd) distinct prime factors from a(54) on, the 54th number of that form being 114141404156 = 2^2 * 13^2 * 19 * 383 * 23203. - M. F. Hasler, Jul 08 2016

			a(1) = 4199030 = 2*5*11*59*647.
a(2) = 29465852 = 2^2*13*23*71*347.
a(3) = 120888092 = 2^2*13*23*61*1657.

M. F. Hasler, Table of n, a(n) for n = 1..60

Cf. A002975, A258401, A258882, A258883, A258885.

select(t -> factor(t)[,2][^1]~ == [1,1,1,1], A002975) \\ Assuming that A002975 is defined as vector holding enough terms of that sequence. - M. F. Hasler, Jul 08 2016

Definition edited (to require p > 2) by M. F. Hasler, Jul 08 2016

Incorrect Mathematica code (allowed 114141404156) removed by M. F. Hasler, Jun 29 2016

A258333 Number of (primitive) weird numbers of the form 2^npq, with odd primes p < q.

Original entry on oeis.org

1, 1, 5, 3, 10, 23, 29, 53, 115, 210, 394, 683, 1389, 3118, 6507, 9120

Offset: 1

Douglas E. Iannucci and Robert G. Wilson v, May 27 2015

Sequence taken from page 3 of "On primitive weird numbers of the form 2^k*p*q".
The (primitive) weird numbers considered here are listed in A258882, a proper subset of A002975.
If 2^k*p*q is weird, then 2^(k+1) < p < 2^(k+2)-2 < q < 2^(2k+1).
This being the case the number of possible pwn of the form 2^n*p*q with p unique is: 1, 2, 4, 7, 12, 23, 43, 75, 137, 255, 463, 872, 1612, 3030, 5708, ....
However, p is usually not unique, e.g., for k=3, p=19 we have two pwn (with q=61 and q=71), and for k=5, p=71 yields two pwn (for q=523 and q=541) and p=67 yields three pwn (for q=887, 971 and 1021). I conjecture that there is an increasing number of pwn with, e.g., p=nextprime(2^(k+1)). Also, if 2^k p q and 2^k p' q are both weird, then usually 2^k p" q is weird for all p" between p and p'. There is one exception [p, p', q] = [2713, 2729, 8191] for k=10, five exceptions [6197, 6203, 12049], [6113, 6131, 12289], [6113, 6131, 12301], [6121, 6133, 12323], [5441, 5449, 16411] for k=11, and seven exceptions for k=12. These exceptions occur when q/p is close to an integer, (p, q) ~ (3/4, 3/2)*2^(k+2) or (2/3, 2)*2^(k+2). - M. F. Hasler, Jul 16 2016

			The only primitive weird number of the form 2*p*q is 70 so a(1) = 1;
The only primitive weird number of the form 2^2*p*q is 836 so a(2) = 1;
There are 5 primitive weird numbers of the form 2^3*p*q and they are 5704, 7912, 9272, 10792 & 17272; so a(3) = 5; etc.

Douglas E. Iannucci, On primitive weird numbers of the form 2^k*p*q, arXiv:1504.02761 [math.NT], 2015.

Cf. A002975, A258882.

A258333(n)={ local(s=0,p,M=2^(n+1)-1,qn,T(P=p-1)=is_A006037(qn*p=precprime(P)) && s+=1); forprime(q=2*M,M*(M+1), qn=q<M, T() || T() || break)); s} \\ Not very efficient, for illustrative purpose only. - M. F. Hasler, Jul 18 2016

a(15) from Robert G. Wilson v, Jun 14 2015

a(16) from Robert G. Wilson v, Dec 06 2015

A258885 Primitive weird numbers (A002975) having 6 distinct prime factors.

Original entry on oeis.org

1550860550, 44257207676, 66072609790

Offset: 1

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

a(4) <= 5976833582079328 = 2^5*181*197*353*431*34429 and a(5) <= 48083019473926272314825065088 = 2^7*257*97213*97973*100957*1520132521 that is certainly in this sequence. - Giuseppe Melfi, Oct 26 2015

a(4) <= 125258675788784 = 2^4 * 47 * 149 * 353 * 1307 * 2423. - M. F. Hasler, Jul 12 2016

			a(1) = 1550860550 = 2 * 5^2 * 29 * 37 * 137 * 211 = A273815(1). (Abundance = 20)
a(2) = 44257207676 = 2^2 * 11 * 37 * 59 * 523 * 881. (Abundance = 8, cf. A088833)
a(3) = 66072609790 = 2 * 5 * 11 * 127^2 * 167 * 223 = A273815(3). (Abundance = 4, cf. A088832)

G. Amato, M. F. Hasler, G. Melfi and M. Parton, Primitive weird numbers having more than three distinct prime factors, Riv. Mat. Univ. Parma, Vol. 7, No. 1 (2016) 153-163.
G. Melfi, On the conditional infiniteness of primitive weird numbers, Journal of Number Theory, 147 (2015) 508-514; Lemma 2.

Cf. A002975, A258401, A258882, A258883, A258884.

(* copy the terms from A002975, assign them to 'lst' and then *)
Select[ lst, PrimeNu@# == 6 &]

select(w->omega(w)==6, A002975) \\ Assuming that A002975 is defined as set or vector. - M. F. Hasler, Jul 12 2016

One more term added and definition corrected by Giuseppe Melfi, Nov 02 2015

A258375 Least primitive weird number with n prime divisors, counting multiplicity.

Original entry on oeis.org

70, 836, 7192, 73616, 519712, 3963968, 33277312, 263144192, 2113834496, 16995175424, 135895635968, 1093862207488, 8752602423296, 70102452125696, 561472495910912, 4494940873621504, 35979456528646144, 287952747599495168

Offset: 3

Robert G. Wilson v, May 28 2015

nonn

A proper subsequence of A002975.
Conjecture: a(n) = the smallest primitive weird number of the form 2^(n-2)*p*q where p*q is minimal.
Is it known that a(n) always exists? - Charles R Greathouse IV, Jun 11 2015
No, it is not even unconditionally proved that there are infinitely many primitive weird numbers. In view of this, the above formula a(n) = 2^(n-2)*p*q and the asymptotic formula a(n) ~ 2^(3n-2) are only conjectures. - M. F. Hasler, Jul 08 2016
The conjectured a(n) ~ 2^(3n-2) follows from the conjecture that a(n) = 2^(n-2)*p*q (cf. A258882) where q is the least prime larger than 2M = 2^n-2 such that 2^(n-2)*q*precprime((Mq-1)/(q-M)) is weird. I also conjecture that for all n > 7, q = nextprime(2^n-2). - M. F. Hasler, Jul 13 2016

			a(3) = 2^1 * 5 * 7 = 70
a(4) = 2^2 * 11 * 19 = 836
a(5) = 2^3 * 29 * 31 = 7192
a(6) = 2^4 * 43 * 107 = 73616
a(7) = 2^5 * 109 * 149 = 519712
a(8) = 2^6 * 241 * 257 = 3963968
a(9) = 2^7 * 499 * 521 = 33277312
a(10) = 2^8 * 997 * 1031 = 263144192
a(11) = 2^9 * 2011 * 2053 = 2113834496
a(12) = 2^10 * 4049 * 4099 = 16995175424
a(13) = 2^11 * 8101 * 8191 = 135895635968
a(14) = 2^12 * 16273 * 16411 = 1093862207488
a(15) = 2^13 * 32603 * 32771 = 8752602423296
a(16) = 2^14 * 65287 * 65537 = 70102452125696
a(17) = 2^15 * 130729 * 131071 = 561472495910912
a(18) = 2^16 * 261637 * 262147 = 4494940873621504
a(19) = 2^17 * 523571 * 524287 = 35979456528646144
a(20) = 2^18 * 1047559 * 1048583 = 287952747599495168
a(21) = 2^19 * 2095721 * 2097169 = 2304288287017664512
a(22) = 2^20 * 4192267 * 4194319 = 18437851191624859648
a(23) = 2^21 * 8385719 * 8388617 = 147523287039340445696
a(24) = 2^22 * 16773149 * 16777259 = 1180308456157336305664
a(25) = 2^23 * 33548689 * 33554467 = 9443126304886073851904
a(26) = 2^24 * 67100681 * 67108879 = 75548667373415913488384
a(27) = 2^25 * 134206169 * 134217757 = 604410983292363190829056
a(28) = 2^26 * 268419077 * 268435459 = 4835408274665227893604352
a(29) = 2^27 * 536847791 * 536870923 = 38683960976635781347016704
a(30) = 2^28 * 1073709061 * 1073741827 = 309475567394195954395512832

M. F. Hasler, Table of n, a(n) for n = 3..30 (first 16 terms from Robert G. Wilson v)
Douglas E. Iannucci, On primitive weird numbers of the form 2^k*p*q, arXiv:1504.02761 [math.NT], 2015.

Cf. A006037, A002975, A258374, A258882.

(* copy the terms from A002975, assign them equal to 'lst' and then *) Table[ Min@ Select[ lst, PrimeOmega@# == n &], {n, 3, 12}]

a(n)=for(k=1,#A=A002975,bigomega(A[k])==n&&return(A[k])) \\ This assumes A002975 is defined as a set or vector with enough terms. A002975 could be replaced by A258882 (for which much larger terms are known) if we assume that all terms are in that sequence. - M. F. Hasler, Jul 08 2016

A258375(n)={ forprime(q=2^n-1,, my(p=precprime((2^(n-1)-1)*(q+1)\(q-2^(n-1)+1)),P); is_A006037(2^(n-2)*p*q) || next; while( is_A006037(2^(n-2)*q*P=precprime(p-1)), p=P); return(2^(n-2)*p*q))} \\ This assumes that all terms are of the form 2^k*p*q. It seems to give correct results at least up to n=30. - M. F. Hasler, Jul 13 2016

It appears that a(n) ~ 2^(3n-2). [Corrected by M. F. Hasler, Jul 13 2016]

a(17) - a(20) from Robert G. Wilson v, Jun 14 2015

a(17) and a(19) corrected, and new terms a(21) - a(30), from M. F. Hasler, Jul 13 2016

A258374 Least primitive weird number with n prime divisors, not counting multiplicity.

Original entry on oeis.org

70, 4030, 4199030, 1550860550

Offset: 3

Robert G. Wilson v, May 28 2015

A proper subset of A002975.
So far all terms are == 2 (mod 4).
From M. F. Hasler, Jul 18 2016: (Start)
a(3) = 2 * 5 * 7 = A002975(1) = A258882(1),
a(4) = 2 * 5 * 13 * 31 = A258401(1) = A258883(1),
a(5) = 2 * 5 * 11 * 59 * 647 = A258884(1),
a(6) = 2 * 5^2 * 29 * 37 * 137 * 211 = A258885(1) = A273815(1). (End)
a(7) <= 4 * 13 * 17 * 449 * 24809 * 228259243 * 11449243661 ≈ 2.6e28, a(8) <= 4 * 13 * 17 * 449 * 24809 * 223842061 * 1123622795959 * 16039588627050434791 ≈ 3.97 e49. - M. F. Hasler, Aug 02 2016
a(7) <= 2 * 5 * 11 * 89 * 167 * 829 * 7972687 ≈ 1.1e16. - M. F. Hasler, Feb 18 2018

Cf. A002975, A258375.

(* copy the terms from A002975, assign them equal to 'lst' and then *) Table[ Min@ Select[ lst, PrimeNu@# == n &], {n, 3, 6}]

A242998 Number of integers k such that R = (2^k*Q - Q - 1)/(Q + 1 - 2^k) is a prime number, when Q = A000668(n) is the n-th Mersenne prime.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 2, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

M. F. Hasler, Aug 17 2014

Related to the search of large primitive weird numbers: Kravitz has shown that 2^(k-1)*Q*R is a primitive weird number (cf. A002975) when Q > 2^k and R = (2^k*Q - Q - 1)/(Q + 1 - 2^k) both are prime. Here we count such primes for the special case where Q = 2^p - 1 is a Mersenne prime, p=A000043(n). For such Q one has R = 2^k - 1 + (2^k - 2)/(2^(p-k) - 1).
See A242025 for the resulting primes R, which however are there not listed in order of the p's.
This sequence gives the row lengths for the table A243003 whose rows hold the k-values leading to prime R, for a given Mersenne prime.

			For given p=A000043(n), the following k's yield a prime R:
p : k's (and resulting primes R, Q=2^p-1 and/or weird W=2^(k-1)*Q*R)
2 : -
3 : 2 (R=5, Q=7, W=70)
5 : 4 (R=29, Q=31, W=7192)
7 : 4 (R=17, Q=127, W=17272), 5 (R=41, Q=127, W=83312)
13 : 11 (R=2729, Q=8191, W=22889716736)
17 : 13 (R=8737, Q=131071, W=4690605371392)
19 : 16 (R=74897, W=1286718208049152), 17 (R=174761, W=6004730783793152)
31 : 16 (R=65537, W=2^15*(2^31-1)*R), 29 (R=715827881, W=2^28*(2^31-1)*R)
61 : 57 (R=153722867280912929, W=2^56*(2^61-1)*R)
89 : 78 (R=302379100949042568368129, W=2^77*(2^89-1)*R)
107 through 86243 : none.
107 through 3021377: none. _Robert Price_, Sep 05 2019
The present sequence lists the number of k's in each line.

S. Kravitz, A search for large weird numbers. J. Recreational Math. 9(1976), 82-85 (1977). Zbl 0365.10003
E. Weisstein, Weird numbers, on MathWorld - a Wolfram web ressource.

Cf. A258882 (PWN of the form 2^k*p*q), A000043 (Mersenne prime exponents), A000668.
Cf. A242025 (the primes R).
Row lengths of A242999 (values of p) and A243003 (values of k), cf. A242993 for the smallest possible k.
See also A320875 for more general solutions to R = (MQ-1)/(Q-M) = prime.

A000043 = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607,
   1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937,
   21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433,
   1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011,
   24036583, 25964951, 30402457, 32582657, 37156667, 42643801,
   43112609};
lst = {};
For[i = 1, i <= 28, i++,
  p = A000043[[i]];
  kc = 0;
  For[k = 1, k < p, k++,
   r = 2^k - 1 + (2^k - 2)/(2^(p - k) - 1);
   If[! IntegerQ[r], Continue[]];
   If[PrimeQ[r], kc++]];
  AppendTo[lst, kc]];
lst (* Robert Price, Sep 05 2019 *)

A242998(n,p=A000043[n])={sum(k=p\2+1, p-1, Mod(2, 2^(p-k)-1)^k==2 && ispseudoprime(2^k-1+(2^k-2)/(2^(p-k)-1)))}

Typo in definition corrected by Jens Kruse Andersen, Aug 27 2014

a(29)-a(37) from Robert Price, Sep 05 2019

cp's OEIS Frontend

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

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A125247 Numbers n whose abundance sigma(n) - 2n = -8. Numbers n whose deficiency is 8.

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A258883 Primitive weird numbers (PWN) of the form 2^k*p*q*r with k > 0 and where p < q < r are odd primes.

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A258401 Primitive weird numbers (A002975) of the form 2^k*p*q*x with k >= 0 and odd p, q, x >= 3.

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A258884 Primitive weird numbers, pwn, of the form 2^k*p*q*r*s with k > 0 and where p < q < r < s are odd primes.

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A258333 Number of (primitive) weird numbers of the form 2^n*p*q, with odd primes p < q.

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A258885 Primitive weird numbers (A002975) having 6 distinct prime factors.

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A258883 Primitive weird numbers (PWN) of the form 2^kpq*r with k > 0 and where p < q < r are odd primes.

A258401 Primitive weird numbers (A002975) of the form 2^kpq*x with k >= 0 and odd p, q, x >= 3.

A258884 Primitive weird numbers, pwn, of the form 2^kpqrs with k > 0 and where p < q < r < s are odd primes.

A258333 Number of (primitive) weird numbers of the form 2^npq, with odd primes p < q.