login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

Table of n, a(n) for n=1..16.

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

Cf. A002975, A258882.

Sequence in context: A141620 A195140 A049829 * A137613 A259650 A165670

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 13 12:45 EST 2019. Contains 329094 sequences. (Running on oeis4.)