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A050922
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Triangle in which n-th row gives prime factors of n-th Fermat number 2^(2^n)+1.
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11
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3, 5, 17, 257, 65537, 641, 6700417, 274177, 67280421310721, 59649589127497217, 5704689200685129054721, 1238926361552897, 93461639715357977769163558199606896584051237541638188580280321
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OFFSET
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0,1
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COMMENTS
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Alternatively, list of prime factors of terms of A001317 in order of their first appearance. - Labos Elemer, Jan 21 2002
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.
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).
This factorization also explains why the "first 31 numbers give odd-sided constructible polygons". I think Hewgill first noticed this factorization. (End)
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REFERENCES
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M. Aigner and G. M. Ziegler, Proofs from The Book, Springer-Verlag, Berlin, 2nd. ed., 2001; see p. 3.
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LINKS
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EXAMPLE
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Triangle begins:
3;
5;
17;
257;
65537;
641, 6700417;
274177, 67280421310721;
59649589127497217, 5704689200685129054721;
1238926361552897, 93461639715357977769163558199606896584051237541638188580280321; ...
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
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MATHEMATICA
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Flatten[Transpose[FactorInteger[#]][[1]]&/@Table[2^(2^n)+1, {n, 0, 8}]] (* Harvey P. Dale, May 18 2012 *)
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PROG
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(PARI) for(n=0, 1e3, f=factor(2^(2^n)+1)[, 1]; for(i=1, #f, print1(f[i], ", "))) \\ Felix Fröhlich, Aug 16 2014
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CROSSREFS
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KEYWORD
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nonn,tabf,nice
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AUTHOR
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Apr 13 2000.
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STATUS
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approved
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