A307843 Divisors of Fermat numbers.

1, 3, 5, 17, 257, 641, 65537, 114689, 274177, 319489, 974849, 2424833, 6700417, 13631489, 26017793, 45592577, 63766529, 167772161, 825753601, 1214251009, 4294967297, 6487031809, 70525124609, 190274191361, 311453532161, 646730219521, 2710954639361

Offset: 1

Author

Jeppe Stig Nielsen, Jul 24 2019

Keywords

• nonn

Comments

Has both A000215 and A023394 as subsequences. Outside these are 1 and the composite proper divisors of Fermat numbers, namely 311453532161, 2983954661377, 7313319444481, ...

Odd m = (p_1)^(e_1)*(p_2)^(e_2)*...*(p_r)^(e_r) is a term if and only if the multiplicative order of 2 modulo (p_i)^(e_i) is the same power of 2 for 1 <= i <= r. - Jianing Song, May 19 2024

Examples

			311453532161 is included because it divides 2^(2^11) + 1. It is not included in A023394 because it is composite.
		

Links

• Jeppe Stig Nielsen, Table of n, a(n) for n = 1..58

Crossrefs

Cf. A000215, A023394, A050922, A094358.

Programs

PARI

isA307843(n) = if(n==1, return(1)); if(n%2, my(f = factor(n), d = znorder(Mod(2,f[1,1]^f[1,2]))); if(!isprimepower(2*d), return(0)); for(i=2, #f~, if(znorder(Mod(2,f[i,1]^f[i,2])) != d, return(0))); 1, 0) \\ Jianing Song, May 19 2024. Inefficient to print the sequence as terms are sparse