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 A258333 Number of (primitive) weird numbers of the form 2^n*p*q, with odd primes p < q. 12
 1, 1, 5, 3, 10, 23, 29, 53, 115, 210, 394, 683, 1389, 3118, 6507, 9120 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS 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 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<M, T() || T() || break)); s} \\ Not very efficient, for illustrative purpose only. - M. F. Hasler, Jul 18 2016 CROSSREFS Cf. A002975, A258882. Sequence in context: A141620 A195140 A049829 * A137613 A335302 A259650 Adjacent sequences:  A258330 A258331 A258332 * A258334 A258335 A258336 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

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Last modified June 22 16:35 EDT 2021. Contains 345388 sequences. (Running on oeis4.)