A050922 - OEIS (original) (raw)

A050922

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

19

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

COMMENTS

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)

REFERENCES

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

EXAMPLE

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

MATHEMATICA

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

PROG

(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