A282572 - OEIS

%I #72 Dec 25 2021 06:56:26

%S 1,3,7,9,15,21,27,31,45,49,63,81,93,105,127,135,147,189,217,225,243,

%T 255,279,315,343,381,405,441,465,511,567,651,675,729,735,765,837,889,

%U 945,961,1023,1029,1143,1215,1323,1395,1519,1533,1575,1701,1785,1905,1953,2025,2047,2187,2205,2295,2401

%N Integers that are a product of Mersenne numbers A000225, (i.e., product of numbers of the form 2^n - 1).

%C Odd orders of finite abelian groups that appear as the group of units in a commutative ring (Chebolu and Lockridge, see A296241). - _Jonathan Sondow_, Dec 15 2017

%C Actually, the Chebolu and Lockridge paper states that this sequence gives all odd numbers that are possible numbers of units in a (commutative or non-commutative) ring (Ditor's theorem). Concretely, if k = (2^(e_1)-1)*(2^(e_2)-1)*...(2^(e_r)-1) is a term, let R = (F_2)^s X F_(2^(e_1)) X F_(2^(e_2)) X ... X F_(2^(e_r)) for s >= 0, then the number of units in R is k. - _Jianing Song_, Dec 23 2021

%H Robert G. Wilson v, <a href="/A282572/b282572.txt">Table of n, a(n) for n = 1..10000</a>

%H Sunil K. Chebolu and Keir Lockridge. <a href="http://arxiv.org/abs/1701.02341">How many units can a commutative ring have?</a>, arXiv preprint arXiv:1701.02341 [math.AC], 2017. See Th. 8.

%e 63 = 1*3^3*7, 81 = 1*3^4, 93 = 1*3*31, 105 = 1*7*15, 41013 = 1*3^3*7^2*31.

%p d:= 15: # for terms < 2^d

%p N:= 2^d:

%p S:= {1}:

%p for m from 2 to d do

%p   r:= 2^m-1;

%p   k:= ilog[r](N);

%p   V:= S;

%p   for i from 1 to k do

%p     V:= select(`<`, map(`*`, V, r), N);

%p     S:= S union V

%p   od;

%p od:

%p sort(convert(S, list)); # _Ridouane Oudra_, Sep 14 2021

%t lmt = 2500; a = b = Array[2^# - 1 &, Floor@ Log2@ lmt]; k = 2; While[k < Length@ a, e = 1; While[e < Floor@ Log[ a[[k]], lmt], b = Union@ Join[b, Select[ a[[k]]^e*b, # < 1 + lmt &]]; e++]; k++]; b (* _Robert G. Wilson v_, Feb 23 2017 *)

%o (PARI) forstep(x=1,1000000,2, t=x; forstep(n=20,2,-1, m=2^n-1; while(t%m==0, t=t\m)); if(t==1, print1(x,","))) \\ _Dmitry Petukhov_, Feb 23 2017

%Y Cf. A000225, A056652, A296241.

%Y Note that A191131, A261524, A261871, and A282572 are very similar and easily confused with each other.

%K nonn

%O 1,2

%A _Andrew Ivashenko_, Feb 18 2017

%E More terms from _Michel Marcus_, Feb 23 2017

%E Definition changed by _David A. Corneth_, Mar 12 2017