A258885 -id:A258885 - 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

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

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}]

A273815 Primitive weird numbers (pwn) with nonsquarefree odd part.

Original entry on oeis.org

1550860550, 2319548096, 66072609790, 114141404156, 232374697216

Offset: 1

M. F. Hasler, Jul 08 2016

Equivalently, primitive weird numbers (A002975) with at least one odd prime factor with multiplicity > 1. A subsequence of A258401.
Although some of them have only few prime divisors, these primitive weird numbers are not in the sequences A258882, A258883, A258884 defined to list "pwn of the form 2^k p*...*r with primes p<... squarefree). Sequence A258885 (pwn with 6 prime divisors) does not have this restriction. - M. F. Hasler, Jul 26 2016
a(6) <= 2^10*2081^2*129083 = 572417848896512, which is also in the sequence. - M. F. Hasler, Feb 15 2018

			a(1) = 1550860550 = 2 * 5^2 * 29 * 37 * 137 * 211 = A258885(1): the smallest pwn with 6 (distinct) prime divisors.
a(2) = 2319548096 = 2^6 * 137^2 * 1931 = A258401(45), but not in A258882 nor A258883, cf. comment.
a(3) = 66072609790 = 2 * 5 * 11 * 127^2 * 167 * 223 = A258885(3).
a(4) = 114141404156 = 2^2 * 13^2 * 19 * 383 * 23203 = A258401(123), but not in A258884, cf. comment.
a(5) = 232374697216 = 2^8 * 797^2 * 1429 = A258401(143), but not in A258882 nor A258883, cf. comment.

Gianluca Amato, Maximilian F. Hasler, Giuseppe Melfi, 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.)

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

select(t->vecmax(factor(t)[,2][^1])>1, A002975) \\ Assuming that A002975 is defined as vector holding enough terms of that sequence

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]

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

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A273815 Primitive weird numbers (pwn) with nonsquarefree odd part.

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

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

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.