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COMMENTS
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Sequence taken from page 3 of "On primitive weird numbers of the form 2^k*p*q".
The (primitive) weird numbers considered here are listed in A258882, a proper subset of A002975.
If 2^k*p*q is weird, then 2^(k+1) < p < 2^(k+2)-2 < q < 2^(2k+1).
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, ....
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
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PROG
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(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
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