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%I #35 Jul 19 2016 09:14:19
%S 1,1,5,3,10,23,29,53,115,210,394,683,1389,3118,6507,9120
%N Number of (primitive) weird numbers of the form 2^n*p*q, with odd primes p < q.
%C Sequence taken from page 3 of "On primitive weird numbers of the form 2^k*p*q".
%C The (primitive) weird numbers considered here are listed in A258882, a proper subset of A002975.
%C If 2^k*p*q is weird, then 2^(k+1) < p < 2^(k+2)-2 < q < 2^(2k+1).
%C 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, ....
%C 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
%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.
%e The only primitive weird number of the form 2*p*q is 70 so a(1) = 1;
%e The only primitive weird number of the form 2^2*p*q is 836 so a(2) = 1;
%e 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.
%o (PARI) 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<<n; T((M*q-1)\(q-M)) || T() || next; while( p>M, T() || T() || break)); s} \\ Not very efficient, for illustrative purpose only. - _M. F. Hasler_, Jul 18 2016
%Y Cf. A002975, A258882.
%K hard,nonn,more
%O 1,3
%A _Douglas E. Iannucci_ and _Robert G. Wilson v_, May 27 2015
%E a(15) from _Robert G. Wilson v_, Jun 14 2015
%E a(16) from _Robert G. Wilson v_, Dec 06 2015
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