OFFSET
1,3
COMMENTS
Sequence taken from page 3 of "On primitive weird numbers of the form 2^k*p*q".
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
LINKS
Douglas E. Iannucci, On primitive weird numbers of the form 2^k*p*q, arXiv:1504.02761 [math.NT], 2015.
EXAMPLE
The only primitive weird number of the form 2*p*q is 70 so a(1) = 1;
The only primitive weird number of the form 2^2*p*q is 836 so a(2) = 1;
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.
PROG
(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
CROSSREFS
KEYWORD
hard,nonn,more
AUTHOR
Douglas E. Iannucci and Robert G. Wilson v, May 27 2015
EXTENSIONS
a(15) from Robert G. Wilson v, Jun 14 2015
a(16) from Robert G. Wilson v, Dec 06 2015
STATUS
approved