[seqfan] Re: A078198 erroneous?

N. J. A. Sloane njas at research.att.com
Thu Mar 18 19:42:47 CET 2010

I just sent the following message to Benoit Cloitre.

>From njas at research.att.com Thu Mar 18 14:32:43 2010
>Date: Thu, 18 Mar 2010 14:32:42 -0400
>From: "N. J. A. Sloane" <njas at research.att.com>
>To: benoit7848c at orange.fr
>Subject: trouble with one of your sequences

>Dear Benoit,  Two days ago there was this sequence in the OEIS:

>%I A078198
>%S A078198 1,0,1,2,3,2,3,4,5,6,7,8,9,10,9,10,11,12,13,14,15,14,15,16,17,18,19,20,
>%T A078198 21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,34,35,36,37,36,37,38,39,
>%U A078198 40,41,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60
>%N A078198 Number of even terms minus number of odd terms in first n terms of A081235.
>%C A078198 Conjecture : a(n)>0 and more precisely : a(n)/sqrt(n) -> + infinity
>%F A078198 Let e(n)=card( 1<=k<=n, A081235(k) is even) and o(n)=card( 1<=k<=n, A081235(k) is odd) then a(n)=e(n)-o(n)
>%K A078198 nonn,new
>%O A078198 1,4
>%A A078198 Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 19 2003

>But we also have

>%I A081235
>%S A081235 2,3,4,4,6,7,8,8,8,10,10,12,12,12,13,14,14,14,16,16,16,17,18,18,18,20
>%N A081235 Erroneous version of A096494.
>%C A081235 I think something has gone wrong here, because this sequence does not look anything like A096494. - N. J. A. Sloane, Mar 18 2010.
>%Y A081235 Sequence in context: A089266 A039696 A076332 this_sequence A092601 A162906 A100476
>%Y A081235 Adjacent sequences: A081232 A081233 A081234 this_sequence A081236 A081237 A081238
>%K A081235 dead
>%O A081235 1,1


>%I A096494
>%S A096494 2,2,4,4,6,6,8,8,8,10,10,12,12,12,12,14,14,14,16,16,16,16,18,18,18,20,
>%T A096494 20,20,20,20,22,22,22,22,24,24,24,24,24,26,26,26,26,26,28,28,28,28,30,
>%U A096494 30,30,30,30,30,32,32,32,32,32,32,32,34,34,34,34,34,36,36,36,36,36,36
>%N A096494 Largest value in period of continued fraction of square root of n-th prime.
>%F A096494 It seems that limit n ->infinity a(n)/n = 0. - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 19 2003
>%F A096494 a(n)=2*A000006(n). - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 19 2003
>%e A096494 n=31,p[31]=127: the period={3,1,2,2,7,11,7,2,2,1,3,22},a[31]=22;
>%t A096494 {te=Table[0, {m}], u=1}; Do[s=Max[Last[ContinuedFraction[Prime[n]^(1/2)]]]; te[[u]]=s;u=u+1, {n, 1, m}];te
>%Y A096494 Cf. A003285, A054269, A005980, A096491-A096496.
>%K A096494 nonn
>%O A096494 1,1
>%A A096494 Labos E. (labos(AT)ana.sote.hu), Jun 29 2004

>So, Benoit, there are errors here, which are probably my fault.

>The question is, what is the correct version of this:

>%N A078198 Number of even terms minus number of odd terms in first n terms of A081235.

>Do you remember?

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