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%I #25 Jul 19 2021 19:56:41
%S 70,4030,5830,4199030,1550860550,66072609790
%N Primitive weird numbers (pwn; A002975) congruent to 2 mod 4.
%C Primitive weird numbers divisible by 2 but not by 4.
%C 10805836895078390 = 2 * 5 * 11 * 89 * 167 * 829 * 7972687 is a term.
%D Gianluca Amato, Maximilian F. Hasler, Giuseppe Melfi, Maurizio Parton. Primitive weird numbers having more than three distinct prime factors. Rivista di Matematica della Università degli studi di Parma, 2016, 7(1), pp. 153-163. (hal-01684543)
%H G. Amato, M. Hasler, G. Melfi and M. 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 Gianluca Amato, Maximilian F. Hasler, Giuseppe Melfi, Maurizio Parton, <a href="https://arxiv.org/abs/1802.07178">Primitive abundant and weird numbers with many prime factors</a>, arXiv:1802.07178 [math.NT], 2018.
%H Stan Benkoski, <a href="http://www.jstor.org/stable/2316276">Problem E2308</a>, Amer. Math. Monthly, 79 (1972) 774.
%H S. J. Benkoski and P. Erdos, <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 R. K. Guy, <a href="/A001599/a001599_1.pdf">Letter to N. J. A. Sloane with attachment, Jun. 1991</a>.
%H 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 Giuseppe 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WeirdNumber.html">Weird Number</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Weird_number">Weird number</a>
%e a(1) is 70 = 2 * 5 * 7 with abundance of 4;
%e a(2) is 4030 = 2 * 5 * 13 * 31 with abundance of 4;
%e a(3) is 5830 = 2 * 5 * 11 * 53 with abundance of 4;
%e a(4) is 4199030 = 2 * 5 * 11 * 59 * 647 with abundance of 20;
%e a(5) is 1550860550 = 2 * 5^2 * 29 * 37 * 137 * 211 with abundance of 20;
%e a(6) is 66072609790 = 2 * 5 * 11 * 127^2 * 167 * 223 with abundance of 4; etc.
%e From _M. F. Hasler_, Nov 28 2018: (Start)
%e The larger terms are in other sequences related to PWN with many prime factors. We have the following relations:
%e a(3) = 70 = A258882(1) = A258374(3) = A258250(1) = A002975(1).
%e a(3) = 4030 = A258883(1) = A258374(4) = A258401(1) = A258250(3) = A002975(3).
%e a(3) = 5830 = A258883(2) = A258401(2) = A258250(4) = A002975(4).
%e a(4) = 4199030 = A258884(1) = A258374(5) = A258401(11) = A265727(15).
%e a(5) = 1550860550 = A258885(1) = A273815(1) = A258374(6).
%e a(6) = 66072609790 = A258885(3) = A273815(3). (End)
%t (* import the b-file in A002975 and assign it to lst *);
%t Select[lst, IntegerExponent[#, 2] == 1 &]
%Y Cf. A002975, A258375, A258401, A258882, A258883, A258884, A258885.
%K nonn,more
%O 1,1
%A _M. F. Hasler_ and _Robert G. Wilson v_, Sep 26 2018
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