This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A002975 M5340 #351 Feb 15 2020 01:25:38
%S A002975 70,836,4030,5830,7192,7912,9272,10792,17272,45356,73616,83312,91388,
%T A002975 113072,243892,254012,338572,343876,388076,519712,539744,555616,
%U A002975 682592,786208,1188256,1229152,1713592,1901728,2081824,2189024,3963968,4128448
%N A002975 Primitive weird numbers: weird numbers with no proper weird divisors.
%C A002975 Sidney Kravitz notes that a(21) = 539744; it was misprinted as 539774 in the Benkoski & Erdős article. - _Charles R Greathouse IV_, Apr 04 2012
%C A002975 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.]
%C A002975 Yes, any primitive weird number, pwn, multiplied by any prime > sigma_1(pwn) is also weird. - _Robert G. Wilson v_, Jun 09 2015
%C A002975 A proper subsequence of A006037 and A091191. - _Robert G. Wilson v_, May 25 2015
%C A002975 Number of terms < 10^n: 0, 1, 2, 7, 13, 24, 48, 85, 152, 276, 499, 881, ..., . - _Robert G. Wilson v_, Jun 21 2017
%C A002975 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
%C A002975 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
%C A002975 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
%C A002975 Sequence A273815 lists terms with nonsquarefree odd part, by definition excluded in A258883 and A258884. - _M. F. Hasler_, Feb 18 2018
%C A002975 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
%C A002975 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
%D A002975 R. K. Guy, Unsolved Problems in Number Theory, B2.
%D A002975 R. Honsberger, Mathematical Gems, M.A.A., 1973, p. 113.
%D A002975 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002975 Robert G. Wilson v, <a href="/A002975/b002975.txt">Table of n, a(n) for n = 1..1161</a> (terms a(58)-a(160) from Donovan Johnson). [Term a(1159) inserted and b-file reformatted by _Georg Fischer_, Jan 16 2019]
%H A002975 Stan Benkoski, <a href="http://www.jstor.org/stable/2316276">Problem E2308</a>, Amer. Math. Monthly, 79 (1972) 774.
%H A002975 Gianluca Amato, Maximilian F. Hasler, Giuseppe Melfi and Maurizio Parton, <a href="http://rivista.math.unipr.it/vols/2016-7-1/amato-et-al.html">Primitive weird numbers having more than three distinct prime factors</a>, Riv. Mat. Univ. Parma, Vol. 7, No. 1 (2016) 153-163.
%H A002975 Gianluca Amato, Maximilian F. Hasler, Giuseppe Melfi and Maurizio Parton, <a href="https://doi.org/10.1016/j.jnt.2019.02.027">Primitive abundant and weird numbers with many prime factors</a>, Journal of Number Theory vol. 201 (2019) pp. 436-459. DOI: 10.1016/j.jnt.2019.02.027. (Preprint: arXiv:1802.07178.)
%H A002975 S. J. Benkoski and P. Erdős, <a href="http://dx.doi.org/10.1090/S0025-5718-1974-0347726-9">On weird and pseudoperfect numbers</a>, Math. Comp., 28 (1974), pp. 617-623. <a href="http://www.renyi.hu/~p_erdos/1974-24.pdf">Alternate link</a>; <a href="http://dx.doi.org/10.1090/S0025-5718-1975-0360452-6">1975 corrigendum</a>.
%H A002975 R. K. Guy, <a href="/A001599/a001599_1.pdf">Letter to N. J. A. Sloane with attachment, Jun. 1991</a>
%H A002975 Douglas E. Iannucci, <a href="http://arxiv.org/abs/1504.02761">On primitive weird numbers of the form 2^k*p*q</a>, arXiv:1504.02761 [math.NT], 2015.
%H A002975 G. Melfi, <a href="http://dx.doi.org/10.1016/j.jnt.2014.07.024">On the conditional infiniteness of primitive weird numbers</a>, Journal of Number Theory, Volume 147, February 2015, Pages 508-514.
%H A002975 Robert G. Wilson v, <a href="/A002975/a002975_23.txt"> Table of n, a(n) along with its abundance and their factorizations for n = 1..1161</a>
%e A002975 10430 = A006037(8) is weird but not primitive weird because it has the proper weird divisor 70 = A006037(1).
%t A002975 (* 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 *)
%t A002975 (* Input: Range of even numbers --- Output: Primitive weird numbers *)
%t A002975 Block[{$RecursionLimit = Infinity},
%t A002975 subOfSum[ss_, kk_, rr_] :=
%t A002975 Module[{s = ss, k = kk, r = rr},
%t A002975 If[s + w[[k]] >= mm && s + w[[k]] <= m, t = False;
%t A002975 Goto[done] (* Found *),
%t A002975 If[s + w[[k]] + w[[k + 1]] <= m,
%t A002975 subOfSum[s + w[[k]], k + 1, r - w[[k]]]];
%t A002975 If[s + r - w[[k]] >= m && s + w[[k + 1]] <= m,
%t A002975 subOfSum[s, k + 1, r - w[[k]] ]]]; t]; (* end subOfSum *)
%t A002975 greedyQ[ab_] := Module[{abn = ab, v, sum, s, j, jj, k}, tt = False;
%t A002975 jj = Length[w]; (* start search *)
%t A002975 Do[s = r; sum = 0; Do[v = w[[j]]; sum = sum + v;
%t A002975 If[sum > abn, sum = sum - v; Goto[nxt]];
%t A002975 If[sum == abn, tt = True; Goto[doneG]]; s = s - v;
%t A002975 Label[nxt], {j, jj, 1, -1}];
%t A002975 jj = jj - 1, {k, 1, jj - 1}]; Label[doneG];
%t A002975 (* True means found, False not found *) tt]; (* end greedyQ *)
%t A002975 cnt = 0;
%t A002975 Do[ If[ Mod[n, 3] == 0, Goto[agn]]; r = DivisorSigma[1, n];
%t A002975 m = r - 2*n;
%t A002975 If[m > 0, fi = FactorInteger[n]; largestP = fi[[Length[fi]]][[1]];
%t A002975 nn = n/largestP; If[m > 2*nn || Length[fi] < 3, Goto[agn]];
%t A002975 If[DivisorSigma[1, nn] > 2*nn, Goto[agn]]; t = True; r = r - n;
%t A002975 ww = Divisors[n]; lenW = Length[ww];
%t A002975 Do[ If[ ww[[i]] <= m, w = Drop[ww, i - lenW]; Break[],
%t A002975 r = r - ww[[i]]], {i, lenW - 1, 1, -1}];
%t A002975 If[r >= m,
%t A002975 If[ greedyQ[m], t = False, (* Powers of 2 dropped *)
%t A002975 exp2 = fi[[1]][[2]]; sig2 = 2^(exp2 + 1) - 1; mm = m - sig2;
%t A002975 lenW = Length[w]; ww = {};
%t A002975 If[exp2 > 1,
%t A002975 Do[ Do[ If[ w[[i]] == 2^ii, ww = AppendTo[ww, w[[i]]]],
%t A002975 {i, 1, lenW}], {ii, 0, exp2}];
%t A002975 w = Complement[w, ww]
%t A002975 (* end T if *), w = Drop[w, 2]];
%t A002975 (* end Pwr2 *) t = subOfSum[0, 1, r]]]; Label[done];
%t A002975 If[t, Print[++cnt, " ", n, " ", t]]];
%t A002975 Label[agn], {n, 2, 10000000, 2}]]
%t A002975 (* from Brent Baughn via http://mathematica.stackexchange.com/questions/73301/calculating-weird-numbers, _Robert G. Wilson v_, Nov 21 2015 *)
%o A002975 (PARI) is_A002975(n)=is_A006037(n)&&!fordiv(n,d,!bittest(d,0)&&d<n&&is_A006037(d)&&return) \\ _M. F. Hasler_, Jan 07 2014
%Y A002975 Cf. A006037.
%Y A002975 Cf. A258882 and A258333; A258883, A258884, A258885 and A258401; A258374 and A258375.
%K A002975 nonn,nice
%O A002975 1,1
%A A002975 _N. J. A. Sloane_
%E A002975 More terms from _Jud McCranie_, Oct 21 2001
%E A002975 One more term from _Robert G. Wilson v_, Aug 04 2009
%E A002975 a(1)-a(123) double-checked by _M. F. Hasler_, Jan 07 2014
%E A002975 Edited by _M. F. Hasler_, Jul 12 2016