(a(1)+a(2)+..a(n)) divides a(1)a(2)..a(n)

Sun Jan 21 18:05:50 CET 2007

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Re: A seqfan forum...
>From: Eric Angelini=20
>Subject: Coprimes and first differences

>Could someone compute a few more terms if this is of interest?
Thanks,
E.

>a(n) and a(n+1) are coprimes, as are first differences d(n) and d(n+1):

>Seq. of conseq. coprimes: 2 5 7 10 17 19 22 27 29 32 37 39 44 47 49 52 ...
>Conseq. coprime diff: 3 2 3  7  2  3  5  2  3  5  2  5  3  2  3 ...

if a(0) =3D 2, a(5) 19 ,

let n =3D 9*k + 5
then :

a(n)=3D  19 + 30*k

a(n+1) =3D a(n) + 3
a(n+2) =3D a(n+1) + 5
a(n+3) =3D a(n+2) + 2
a(n+4) =3D a(n+3) + 3
a(n+5) =3D a(n+4) + 5
a(n+6) =3D a(n+5) + 2
a(n+7) =3D a(n+6) + 5
a(n+8) =3D a(n+7) + 3

a(n+9) =3D 19 + 30*(k+1) =3D a(n)+30
etc.

Regards
JT

for example k=3D222 .  a(2003) =3D 6679, a(2007) =3D 6692

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<HTML><HEAD><TITLE>Re: A seqfan forum...</TITLE>
<META http-equiv=3DContent-Type content=3D"text/html; charset=3Diso-8859-1">
<META content=3D"MSHTML 6.00.2900.3020" name=3DGENERATOR>
<STYLE></STYLE>
</HEAD>
<BODY bgColor=3D#ffffff>
<DIV><FONT face=3DArial size=3D2></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2>>From: Eric Angelini <BR>>Subject: C=
oprimes=20
and first differences</FONT></DIV>
<DIV> </DIV>
<DIV><FONT face=3DArial size=3D2></FONT> </DIV>
<DIV> </DIV>
<DIV><FONT face=3DArial size=3D2>>Could someone compute a few more terms=
 if this=20
is of interest?<BR>Thanks,<BR>E.</FONT></DIV>
<DIV> </DIV>
<DIV><FONT face=3DArial size=3D2>>a(n) and a(n+1) are coprimes, as are f=
irst=20
differences d(n) and d(n+1):</FONT></DIV>
<DIV> </DIV>
<DIV><FONT face=3DArial size=3D2>>Seq. of conseq. coprimes: 2 5 7 10 17 =
19 22 27=20
29 32 37 39 44 47 49 52 ...<BR>>Conseq. coprime diff: 3 2 3  7&nbsp=
;=20
2  3  5  2  3  5  2  5  3  2&n=
bsp;=20
3 ...</FONT></DIV>
<DIV> </DIV>
<DIV><FONT face=3DArial size=3D2>if a(0) =3D 2, a(5) 19 ,</FONT></DIV>
<DIV><FONT face=3DArial size=3D2><BR>let n =3D 9*k + 5<BR>then :</DIV>
<DIV> </DIV>
<DIV>a(n)=3D  19 + 30*k</DIV>
<DIV> </DIV>
<DIV>a(n+1) =3D a(n) + 3<BR>a(n+2) =3D a(n+1) + 5<BR>a(n+3) =3D a(n+2) + 2<=
BR>a(n+4) =3D=20
a(n+3) + 3<BR>a(n+5) =3D a(n+4) + 5<BR>a(n+6) =3D a(n+5) + 2<BR>a(n+7) =3D =
a(n+6) +=20
5<BR>a(n+8) =3D a(n+7) + 3</DIV>
<DIV> </DIV>
<DIV>a(n+9) =3D 19 + 30*(k+1) =3D a(n)+30<BR>etc.</DIV>
<DIV> </DIV>
<DIV>Regards<BR>JT</DIV>
<DIV> </DIV>
<DIV>for example k=3D222 .  a(2003) =3D 6679, a(2007) =3D 6692</DIV>
<DIV> </DIV>
<DIV><BR></FONT> </DIV></BODY></HTML>

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--Apple-Mail-1--18043918

Leroy Quet:

> Is the a(1)=3 sequence the same as the a(1)=1,a(2)=2 sequence after  
> some point?

Yes. The sorted list of the first 103 terms of one equals the sorted  
list of the first 103 terms of the other. So, beginning with term  
#104, both sequences are {... 104, 105, 108, 107, 110, 113, 115, ...}.
--Apple-Mail-1--18043918

<HTML><BODY style=3D"word-wrap: break-word; -khtml-nbsp-mode: space; -khtml=
pple-interchange-newline"><BLOCKQUOTE type=3D"cite"><P style=3D"margin: 0.0=
px 0.0px 0.0px 0.0px"><FONT face=3D"Monaco" size=3D"5" style=3D"font: 16.0p=
x Monaco">Is the a(1)=3D3 sequence the same as the a(1)=3D1,a(2)=3D2 sequen=
ce after some<SPAN class=3D"Apple-converted-space">=A0point?</SPAN></FONT><=
/P> </BLOCKQUOTE></DIV><BR><DIV>Yes. The sorted list of the first 103 terms=
inning with term #104, both sequences are {...=A0104, 105, 108, 107, 110, 1=
13, 115, ...}.</DIV></BODY></HTML>
--Apple-Mail-1--18043918--

(a(1)+a(2)+..a(n)) divides a(1)*a(2)*..a(n)

(a(1)+a(2)+..a(n)) divides a(1)a(2)..a(n)