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%I #71 May 04 2019 18:28:38
%S 70,836,7192,7912,9272,10792,17272,73616,83312,113072,519712,539744,
%T 555616,682592,786208,1188256,1229152,1901728,2081824,2189024,3963968,
%U 4128448,4145216,4486208,4559552,4632896,4960448,5440192,5568448,6460864,6621632,7354304,7470272,8000704,8134208
%N Primitive weird numbers of the form 2^k*p*q with k > 0 and where p < q are odd primes.
%C The number of terms < 10^n: 0, 1, 2, 5, 9, 15, 35, 61, 114, 204, 380, 696, 1703, 3548, 6726, 13137, ....
%C 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).
%C 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
%C 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
%C 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
%C 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
%D S. Kravitz, A search for large weird numbers. J. Recreational Math. 9 (1976), 82-85 (1977). Zbl 0365.10003
%H Douglas E. Iannucci and Robert G. Wilson v, <a href="/A258882/b258882.txt">Table of n, a(n) for n = 1..15384</a>, updated Dec 06 2015; corrected by _M. F. Hasler_, Jul 16 2016
%H R. Bagula et al., <a href="http://web.archive.org/web/20150913004131/https://www.linkedin.com/grp/post/4510047-5820529955682926595">A very big weird number</a>, Number Theory group on LinkedIn (web.archive.org snapshot; page no longer available). Dec. 2013
%H Central Washington University, <a href="http://www.cwu.edu/cwu-math-students-break-world-record-largest-weird-number-0">CWU Math Students Break World Record for Largest Weird Number</a> [alternate article]
%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, Vol. 147, Feb 2015, pp 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>
%F A258882 union A258401 is A002975.
%e a(1) = A002975(1) = 70 = 2*5*7.
%e a(2) = A002975(2) = 836 = 2^2*11*19.
%e A002975(3) = 4030 = 2*5*13*31 is not in this sequence since it is not of the required form.
%e The same is true for A002975(4) = 5830.
%e a(3) = A002975(5) = 7192 = 2^3*29*31, etc.
%e 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
%t (* 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]
%o (PARI) select(t->factor(t)[, 2][^1]=[1, 1]~, A002975) \\ Assuming that A002975 is defined as set or vector. - _M. F. Hasler_, Jul 11 2016
%Y Cf. A002975, A258401 (PWN not of this form), A258374, A258375, A258883, A258884, A258885.
%Y 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).
%Y Cf. A320875 (more general case of Karavitz' formula).
%K nonn
%O 1,1
%A _Douglas E. Iannucci_ and _Robert G. Wilson v_, Jun 14 2015
%E Edited by _M. F. Hasler_, Jul 11 2016, Nov 20 2018
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