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%I #57 Apr 03 2023 10:36:09
%S 3,5,17,257,65537,641,6700417,274177,67280421310721,59649589127497217,
%T 5704689200685129054721,1238926361552897,
%U 93461639715357977769163558199606896584051237541638188580280321
%N Triangle in which n-th row gives prime factors of n-th Fermat number 2^(2^n)+1.
%C Alternatively, list of prime factors of terms of A001317 in order of their first appearance. - _Labos Elemer_, Jan 21 2002
%C From _T. D. Noe_, Jan 29 2009: (Start)
%C 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 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 This factorization also explains why the "first 31 numbers give odd-sided constructible polygons". I think Hewgill first noticed this factorization. (End)
%D M. Aigner and G. M. Ziegler, Proofs from The Book, Springer-Verlag, Berlin, 2nd. ed., 2001; see p. 3.
%H Jeppe Stig Nielsen, <a href="/A050922/b050922.txt">Table of n, a(n) for n = 0..29</a>
%H J. Bernheiden, <a href="http://www.mathe-schule.de/download/pdf/Primzahl/Fermat.pdf">Fermat Numbers (Text in German)</a>
%H R. P. Brent, <a href="https://maths-people.anu.edu.au//~brent/pub/pub161.html">Factorization of the tenth Fermat number</a>
%H R. P. Brent, <a href="https://maths-people.anu.edu.au/~brent/pub/pub113.html">Factorization of the eleventh Fermat number</a>
%H R. P. Brent, <a href="https://maths-people.anu.edu.au/~brent/pub/pub066.html">Succinct proofs of primality for the factors of some Fermat numbers</a>
%H R. P. Brent & J. M. Pollard, <a href="https://maths-people.anu.edu.au/~brent/pub/pub061.html">Factorization of the eighth Fermat number</a>
%H R. P. Brent et al., <a href="https://maths-people.anu.edu.au/~brent/pub/pub175.html">Three new factors of Fermat numbers</a>
%H C. K. Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/page.php/FermatDivisor.html">Fermat divisor</a>
%H Wilfrid Keller, <a href="http://www.prothsearch.com/fermat.html">Prime factors k.2^n + 1 of Fermat numbers F_m</a>
%H R. Mestrovic, <a href="http://arxiv.org/abs/1202.3670">Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof</a>, arXiv preprint arXiv:1202.3670, 2012. - From _N. J. A. Sloane_, Jun 13 2012
%H R. Munafo, <a href="http://www.mrob.com/pub/math/ln-notes1.html#fermat">Notes on Fermat numbers</a>
%H Mercedes Orús-Lacort, <a href="https://doi.org/10.13140/RG.2.2.31221.73449">Fermat numbers are not prime numbers for n >= 5</a>, (2020).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FermatNumber.html">Fermat Number</a>
%e Triangle begins:
%e 3;
%e 5;
%e 17;
%e 257;
%e 65537;
%e 641, 6700417;
%e 274177, 67280421310721;
%e 59649589127497217, 5704689200685129054721;
%e 1238926361552897, 93461639715357977769163558199606896584051237541638188580280321; ...
%e 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 Elemer_, Jan 21 2002
%t Flatten[Transpose[FactorInteger[#]][[1]]&/@Table[2^(2^n)+1,{n,0,8}]] (* _Harvey P. Dale_, May 18 2012 *)
%o (PARI) for(n=0, 1e3, f=factor(2^(2^n)+1)[, 1]; for(i=1, #f, print1(f[i], ", "))) \\ _Felix Fröhlich_, Aug 16 2014
%Y Cf. A000215, A019434, A023394, A093179.
%Y Cf. A001317, A001316, A003401, A045544, A053576.
%K nonn,tabf,nice
%O 0,1
%A _N. J. A. Sloane_, Dec 30 1999
%E More terms from Larry Reeves (larryr(AT)acm.org), Apr 13 2000.
%E Edited by _N. J. A. Sloane_, Jan 31 2009 at the suggestion of _T. D. Noe_
%E Link to Munafo webpage fixed by _Robert Munafo_, Dec 09 2009
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