EDITED A014663

[Maximilian Hasler]

sorry for the glitch in the PARI code, must be:
isA014663(p)=1==... as below and not ... 1 != ... as in the previous mail!

%I A014663
%S A014663 7,23,31,47,71,73,79,89,103,127,151,167,191,199,223,233,
%T A014663 239,263,271,311,337,359,367,383,431,439,463,479,487,503,
%U A014663 599,601,607,631,647,719,727,743,751,823,839,863,881,887
%N A014663 Primes p such that order of 2 mod p is odd.
%C A014663 Or, primes p which do not divide 2^n+1 for any n.
%C A014663 The possibility n=0 in the above rules out A072936(1)=2;
apart from this, A014663(n)=A072936(n+1). - M. Hasler, Dec 08 2007
%C A014663 The order of 2 mod p is odd iff 2^k=1 mod p, where
p-1=2^s*k, k odd. - M. Hasler, Dec 08 2007
%D A014663 H. H. Hasse, Ueber die Dichte der Primzahlen p, ..., Math.
Ann., 168 (1966), 19-23.
%D A014663 L. C. Lagarias, The set of primes dividing the Lucas
numbers has density 2/3, Pacific
               J. Math., 118 (1985), 449-461.
%D A014663 P. Moree, Appendix to V. Pless et al., Cyclic Self-Dual Z_4
Codes, Finite Fields
               Applic., vol. 3 pp. 48-69, 1997.
%H A014663 T. D. Noe, <a
href="http://www.research.att.com/~njas/sequences/b014663.txt">Table
               of n, a(n) for n=1..1000</a>
%F A014663 Has density 7/24 (Hasse)
%o A014663 (PARI) isA014663(p)=1==Mod(1,p)<<((p-1)>>factor(p-1,2)[1,2])
listA014663(N=1000)=forprime(p=3,N,isA014663(p)&print1(p", ")) \\ - M.
Hasler, Dec 08 2007
%Y A014663 Complement in primes of A091317.
%Y A014663 Cf. A040098, A045315, A049564.
%Y A014663 Essentially the same as A072936 (except for missing leading term 2).
%Y A014663 Adjacent sequences: A014660 A014661 A014662 this_sequence
A014664 A014665 A014666
%Y A014663 Sequence in context: A036259 A004628 A089199 this_sequence
A007522 A098029 A098039
%K A014663 nonn
%O A014663 1,1
%A A014663 njas
%E A014663 Edited by Maximilian F. Hasler
(maximilian.hasler(AT)gmail.com), Dec 08 2007




mh> From seqfan-owner at ext.jussieu.fr  Sat Dec  8 14:28:40 2007
mh> Date: Sat, 8 Dec 2007 09:28:26 -0400
mh> From: "Maximilian Hasler" <maximilian.hasler at gmail.com>
mh> To: Seqfan <seqfan at ext.jussieu.fr>, "N. J. A. Sloane" <njas at research.att.com>
mh> Subject: primes that never divide 2^k+1
mh> 
mh> Dear seqfans,
mh> There is a series of sequences of primes which are almost the same:
mh> primes which never divide 2^n+1, primes = {2,7} mod 8, and
mh> primes such that x^(...) = 2 has a solution mod p :
mh> 
mh> A045315 A072936 A049564 A049584 A045382 A049560 etc etc
mh> http://www.research.att.com/~njas/sequences/?q=7,23,31,47,71,73,79,89&fmt=1
mh> 
mh> some of them coincide for all terms that are listed, e.g.
mh> http://www.research.att.com/~njas/sequences/?q=id:A049564|id:A072936
mh> agree for all terms and there is no comment concerning that issue.
mh> - BTW,  the ..64 one has offset =0, could be fixed at that occasion).
mh> I think it would be useful to add cross references, comments on the differences
mh> and/or  subsequence relations among them.
mh> (I think those "x^(...) = 2 has a solution" are subsequences of
mh> A072936 differing in only few terms, so it would be more interesting
mh> to know what terms of A072936 are NOT in them)
mh> Regards,
mh> Maximilian
mh> PS: and A014663 is essentially the same than A072936 ; b.t.w. the
mh> comment should be amended to "ODD primes...", I'll send an EDIT for
mh> that one.


The smallest solutions x associated with the sequence are 0, 2, 8, 15, 17, 15,
18, 3, 5, 48 etc in that order,  and a subsequence of these builds A065907.

Members of this sequence which are not in A014663 are
2, 113, 257, 281, 353, 577, 593, 617, 1033, 1049, 1097 etc. They have smallest

Members of this sequence which are not in A045315 are 113, 281, 353, 577, 593,
617, 1033, 1049, 1097 etc.

--
Second proposal: add the following three comments to A045315:

The smallest solutions x associated with the sequence are 0, 3, 10, 4, 8, 21,
5, 32, 9, 40, 2, 4 etc in that order.

Members of this sequence which are not in A014663 are
2, 257, 1217, 1249, 1553, 1777, 2113, 2657, 2833, 4049 etc. Their associated
list of solutions x is 0, 42, 82, 88, 107, 240, 106, 129 etc.

Members of this sequence which are not in A049564 are listed in A070184.

--
Richard