login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A094358 Squarefree products of factors of Fermat numbers (A023394). 11
1, 3, 5, 15, 17, 51, 85, 255, 257, 641, 771, 1285, 1923, 3205, 3855, 4369, 9615, 10897, 13107, 21845, 32691, 54485, 65535, 65537, 114689, 163455, 164737, 196611, 274177, 319489, 327685, 344067, 494211, 573445, 822531, 823685, 958467, 974849, 983055 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

641 is the first member not in sequences A001317, A004729, etc.

Conjectured (by Munafo, see link) to be the same as: numbers n such that 2^^n == 1 mod n, where 2^^n is A014221(n).

It is clear from the observations by Max Alekseyev in A023394 and the Chinese remainder theorem that any squarefree product b of divisors of Fermat numbers satisfies 2^(2^b) == 1 (mod b), hence satisfies Munafo's congruence above. The converse is true iff all Fermat numbers are squarefree. However, if nonsquarefree Fermat numbers exist, the criterion that is equivalent with Munafo's property would be "numbers b such that each prime power that divides b also divides some Fermat number". - Jeppe Stig Nielsen, Mar 05 2014

Also numbers b such that b is (squarefree and) a divisor of A051179(m) for some m. Or odd (squarefree) b where the multiplicative order of 2 mod b is a power of two. - Jeppe Stig Nielsen, Mar 07 2014

LINKS

Robert G. Wilson, T. D. Noe and Ray Chandler, Table of n, a(n) for n = 1..3393 (Original 55 terms from Robert G. Wilson, extended to 1314 terms by T. D. Noe)

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

Robert Munafo, Sequence A094358, 2^^A(N) = 1 mod N

EXAMPLE

3 is a member because it is in A023394. 51 is a member because it is 3*17 and 17 is also in A023394. 153=3*3*17 is not a member because its factorization includes two 3's.

See the Munafo link for examples of the (conjectured) 2^^n == 1 mod n property.

PROG

(PARI) (  isOK1(n) = n%2==1 && hammingweight(znorder(Mod(2, n)))==1  ) ;  (  isOK2(n) = issquarefree(n) && isOK1(n)  )  \\ isOK1 and isOK2 differ only if n contains a prime square that divides a Fermat number (none are known), Jeppe Stig Nielsen, Apr 02 2014

CROSSREFS

Cf. A023394, A014221, A092188, A001317, A004729.

Sequence in context: A071593 A192794 A018358 * A003527 A004729 A045544

Adjacent sequences:  A094355 A094356 A094357 * A094359 A094360 A094361

KEYWORD

nonn

AUTHOR

Robert Munafo, Apr 26 2004

EXTENSIONS

Edited by T. D. Noe, Feb 02 2009

Example brought in line with name/description by Robert Munafo, May 18 2011

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified January 21 14:50 EST 2017. Contains 281109 sequences.