A282572 Integers that are a product of Mersenne numbers A000225, (i.e., product of numbers of the form 2^n - 1).

1, 3, 7, 9, 15, 21, 27, 31, 45, 49, 63, 81, 93, 105, 127, 135, 147, 189, 217, 225, 243, 255, 279, 315, 343, 381, 405, 441, 465, 511, 567, 651, 675, 729, 735, 765, 837, 889, 945, 961, 1023, 1029, 1143, 1215, 1323, 1395, 1519, 1533, 1575, 1701, 1785, 1905, 1953, 2025, 2047, 2187, 2205, 2295, 2401

Offset: 1

Author

Andrew Ivashenko, Feb 18 2017

Keywords

• nonn

Comments

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

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

Examples

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

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..10000
• Sunil K. Chebolu and Keir Lockridge. How many units can a commutative ring have?, arXiv preprint arXiv:1701.02341 [math.AC], 2017. See Th. 8.

Crossrefs

Cf. A000225, A056652, A296241.

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

Programs

Maple

d:= 15: # for terms < 2^d
N:= 2^d:
S:= {1}:
for m from 2 to d do
  r:= 2^m-1;
  k:= ilog[r](N);
  V:= S;
  for i from 1 to k do
    V:= select(`<`, map(`*`, V, r), N);
    S:= S union V
  od;
od:
sort(convert(S, list)); # Ridouane Oudra, Sep 14 2021

Mathematica

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 *)

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

Extensions

More terms from Michel Marcus, Feb 23 2017

Definition changed by David A. Corneth, Mar 12 2017