Search: id:A050922
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%I A050922
%S A050922 3,5,17,257,65537,641,6700417,274177,67280421310721,59649589127497217,
%T A050922 5704689200685129054721,1238926361552897,
%U A050922 93461639715357977769163558199606896584051237541638188580280321
%N A050922 Triangle in which n-th row gives prime factors of n-th Fermat number 
               2^(2^n)+1.
%C A050922 Alternatively, list of prime factors of terms of A001317 in order of 
               their first appearance. - Labos E. (labos(AT)ana.sote.hu), Jan 21 
               2002
%C A050922 Comments from T. D. Noe, Jan 29 2009 (Start): That these two definitions 
               give the same sequence follows from the fact (stated as a formula 
               in A001317) that A001317(n) is the product of Fermat numbers F(i) 
               according to which bits of n are set.
%C A050922 For instance, for n=41, the binary representation of n is 101001, which 
               has bits 0, 3 and 5 set. A001317(n) = 3311419785987 = 3*257*4294967297 
               = F(0)*F(3)*F(5).
%C A050922 This factorization also explains why the "first 31 numbers give odd-sided 
               constructible polygons". I think Hewgill first noticed this factorization. 
               (End)
%D A050922 M. Aigner and G. M. Ziegler, Proofs from The Book, Springer-Verlag, Berlin, 
               2nd. ed., 2001; see p. 3.
%H A050922 J. Bernheiden, <a href="http://www.mathe-schule.de/download/pdf/Primzahl/
               Fermat.pdf">Fermat Numbers (Text in German)</a>
%H A050922 R. P. Brent, <a href="http://wwwmaths.anu.edu.au/~brent/pub/pub161.html">
               Factorization of the tenth Fermat number</a>
%H A050922 R. P. Brent, <a href="http://wwwmaths.anu.edu.au/~brent/pub/pub113.html">
               Factorization of the eleventh Fermat number</a>
%H A050922 R. P. Brent, <a href="http://wwwmaths.anu.edu.au/~brent/pub/pub066.html">
               Succint proofs of primality for the factors of some Fermat numbers</
               a>
%H A050922 R. P. Brent & J. M. Pollard, <a href="http://wwwmaths.anu.edu.au/~brent/
               pub/pub061.html">Factorization of the eighth Fermat number</a>
%H A050922 R. P. Brent et al., <a href="http://wwwmaths.anu.edu.au/~brent/pub/pub175.html">
               Three new factors of Fermat numbers</a>
%H A050922 C. K. Caldwell, The Prime Glossary, <a href="http://primes.utm.edu/glossary/
               page.php/FermatDivisor.html">Fermat divisor</a>
%H A050922 Wilfrid Keller, <a href="http://www.prothsearch.net/fermat.html">Prime 
               factors k.2^n + 1 of Fermat numbers F_m</a>
%H A050922 R. Munafo, <a href="http://home.earthlink.net/~mrob/pub/math/ln-notes1.html#fermat">
               Notes on Fermat numbers</a>
%H A050922 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               FermatNumber.html">Fermat Number</a>
%e A050922 Triangle begins:
%e A050922 3;
%e A050922 5;
%e A050922 17;
%e A050922 257;
%e A050922 65537;
%e A050922 641, 6700417;
%e A050922 274177, 67280421310721;
%e A050922 59649589127497217, 5704689200685129054721;
%e A050922 1238926361552897, 93461639715357977769163558199606896584051237541638188580280321; 
               ...
%e A050922 A001317(127) = 3.5.17.257.65537.641.6700417.274177.6728042130721, A001317(128) 
               = 59649589127497217.5704689200685129054721. See also A050922. Compare 
               with A053576, where 2 and A000215 appear as prime factors. - Labos 
               E. (labos(AT)ana.sote.hu), Jan 21 2002
%Y A050922 Cf. A000215, A019434, A093179.
%Y A050922 Cf. A001317, A001316, A003401, A045544, A053576, A050922.
%Y A050922 Sequence in context: A125045 A093179 A067387 this_sequence A070592 A000215 
               A123599
%Y A050922 Adjacent sequences: A050919 A050920 A050921 this_sequence A050923 A050924 
               A050925
%K A050922 nonn,tabf,nice
%O A050922 0,1
%A A050922 N. J. A. Sloane (njas(AT)research.att.com), Dec 30 1999
%E A050922 More terms from Larry Reeves (larryr(AT)acm.org), Apr 13 2000.
%E A050922 Edited by N. J. A. Sloane (njas(AT)research.att.com), Jan 31 2009 at 
               the suggestion of T. D. Noe

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