A050922 Triangle in which n-th row gives prime factors of n-th Fermat number 2^(2^n)+1.

3, 5, 17, 257, 65537, 641, 6700417, 274177, 67280421310721, 59649589127497217, 5704689200685129054721, 1238926361552897, 93461639715357977769163558199606896584051237541638188580280321

Offset: 0

Author

N. J. A. Sloane, Dec 30 1999

Keywords

• nonn
• tabf
• nice

Comments

Alternatively, list of prime factors of terms of A001317 in order of their first appearance. - Labos Elemer, Jan 21 2002

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.

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)

Examples

			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
		

References

• M. Aigner and G. M. Ziegler, Proofs from The Book, Springer-Verlag, Berlin, 2nd. ed., 2001; see p. 3.

Links

• Jeppe Stig Nielsen, Table of n, a(n) for n = 0..29
• J. Bernheiden, Fermat Numbers (Text in German)
• R. P. Brent, Factorization of the tenth Fermat number
• R. P. Brent, Factorization of the eleventh Fermat number
• R. P. Brent, Succinct proofs of primality for the factors of some Fermat numbers
• R. P. Brent & J. M. Pollard, Factorization of the eighth Fermat number
• R. P. Brent et al., Three new factors of Fermat numbers
• C. K. Caldwell, The Prime Glossary, Fermat divisor
• Wilfrid Keller, Prime factors k.2^n + 1 of Fermat numbers F_m
• R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670, 2012. - From _N. J. A. Sloane_, Jun 13 2012
• R. Munafo, Notes on Fermat numbers
• Mercedes Orús-Lacort, Fermat numbers are not prime numbers for n >= 5, (2020).
• Eric Weisstein's World of Mathematics, Fermat Number

Crossrefs

Cf. A000215, A019434, A023394, A093179.

Cf. A001317, A001316, A003401, A045544, A053576.

Programs

Mathematica

Flatten[Transpose[FactorInteger[#]][[1]]&/@Table[2^(2^n)+1,{n,0,8}]] (* Harvey P. Dale, May 18 2012 *)

PARI

for(n=0, 1e3, f=factor(2^(2^n)+1)[, 1]; for(i=1, #f, print1(f[i], ", "))) \\ Felix Fröhlich, Aug 16 2014

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Apr 13 2000.

Edited by N. J. A. Sloane, Jan 31 2009 at the suggestion of T. D. Noe

Link to Munafo webpage fixed by Robert Munafo, Dec 09 2009