

A094358


Squarefree products of factors of Fermat numbers (A023394).


12



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
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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 v, 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 from T. D. Noe)
Sourangshu Ghosh and Pranjal Jain, On Fermat Numbers and Munafo's Conjecture, (2021).
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 20122018.  From N. J. A. Sloane, Jun 13 2012
Robert Munafo, Sequence A094358, 2^^A(N) = 1 mod N


EXAMPLE

3 is a term because it is in A023394.
51 is a term because it is 3*17 and 17 is also in A023394.
153 = 3*3*17 is not a term because its factorization includes two 3's.
See the Munafo link for examples of the (conjectured) 2^^n == 1 (mod n) property.


MATHEMATICA

kmax = 10^6;
A023394 = Select[Prime[Range[kmax]], IntegerQ[Log[2, MultiplicativeOrder[2, #] ] ]&];
Reap[For[k = 1, k <= kmax, k++, ff = FactorInteger[k]; If[k == 1  AllTrue[ff, MemberQ[A023394, #[[1]]] && #[[2]] == 1 &], Print[k]; Sow[k]]]][[2, 1]] (* JeanFrançois Alcover, Nov 03 2018 *)


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 squared prime that divides a Fermat number (none are known) \\ Jeppe Stig Nielsen, Apr 02 2014


CROSSREFS

Cf. A023394, A014221, A092188, A001317, A004729.
Sequence in context: A293001 A018358 A319583 * 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



