AW: A question about A079490

[Pfoertner, Hugo]

Mark,

I have only 33 digits accuracy with VAX quadruple precision, but that is
sufficient
to exclude 62 from the sequence. With
C A079490
      INTEGER ARG
      REAL*16 QARG, EX, DIFF, DIFMIN
      DIFMIN = 100.0Q0
      DO 10 ARG = 1, 76
      QARG = QEXT ( ARG )
      EX = QEXP(QARG)
      DIFF = EX - QNINT(EX)
      IF ( QABS(DIFF) .LT. DIFMIN ) THEN
        DIFMIN = QABS(DIFF)
        WRITE (*,*) ARG, DIFF
      ENDIF
10    CONTINUE
      END

I get:
          1 -0.281718171540954764639712528647338
          3  8.553692318766774092852965458171901E-0002
          8 -4.201295827172525640790054711137231E-0002
         19 -3.681273915508996621127729387694541E-0002
         45  3.479723340609908177611941937357187E-0002
         75  0.000000000000000000000000000000000E+0000

on my Digital Alphastation

Hugo Pfoertner
http://www.pfoertner.org/

-----Ursprüngliche Nachricht-----
Von: Mark Hudson [mailto:mrmarkhudson at hotmail.com]
Gesendet am: 22 January, 2003 13:14
An: seqfan at ext.jussieu.fr
Betreff: A question about A079490

Hi all,

I have a question about sequence A079490.

ID Number: A079490
Sequence:  1,3,8,19,45,62
Name:      Exp(n) is closer to an integer than any previous exp(k) for 1 <= 
k < n.
Example:   a(2) = 3, exp(3) = 20.08... is closer to an integer than exp(1) =

2.718...
Program:   (PARI) d=1; for( n=1,500, x=exp(n); y=1/(( x-floor(x)) *
              (ceil(x)-x)) ; if( y>d, print( n); d= y, ))

I was thinking of extending the sequence when I ran into a question.

The terms I get are:

1,3,8,19,45,75,135,178,209,732,...

so my sequence diverges from the one in the database after term=45.

If my understanding of the definition of the sequence is correct, then I 
should look at the fractional parts of exp(n) for each term:

frac(exp(45)) = 0.03479...
frac(exp(62)) = 0.0373...
frac(exp(75)) = 0.01434...

Although I may have missed a term before 75 which is closer to its nearest 
integer than 45 is, since exp(62)) is further from an integer than exp(45) 
is, I do not see that this can be the next term in the sequence after 45.

Mark.