A296241 -id:A296241 - cp's OEIS frontend

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

Andrew Ivashenko, Feb 18 2017

nonn

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

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

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.

Cf. A000225, A056652, A296241.

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

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

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 *)

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

More terms from Michel Marcus, Feb 23 2017

Definition changed by David A. Corneth, Mar 12 2017

A070932 Possible number of units in a finite (commutative or non-commutative) ring.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 31, 32, 36, 40, 42, 44, 45, 46, 48, 49, 52, 54, 56, 58, 60, 62, 63, 64, 66, 70, 72, 78, 80, 81, 82, 84, 88, 90, 92, 93, 96, 98, 100

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), May 24 2002

This is a list of the numbers of units in R where R ranges over all finite commutative or non-commutative rings.
By considering the ring Z_n and the finite fields GF(q) this sequence contains the values of the Euler function phi(n) (A000010) and prime powers - 1 (A181062). By taking direct product of rings, if n and m belong to the sequence then so does m*n.
Eric M. Rains has shown that these rules generate all terms of this sequence. More precisely, he shows this sequence (with 0 removed) is the multiplicative monoid generated by all numbers of the form q^n-q^{n-1} for n >= 1 and q a prime power (see Rains link).
Since the number of units of F_q[X]/(X^n) is q^n - q^(n-1), restricting to finite commutative rings gives the same sequence. A296241, which is a proper supersequence, allows the ring R to be infinite. - Jianing Song, Dec 24 2021

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
E. M. Rains, Comments on A070932

Cf. A000010, A002202, A000252, A000961, A181062, A221178.
A000252 is a subsequence.
A282572 is the subsequence of odd terms.
Proper subsequence of A296241.
The main entries concerned with the enumeration of rings are A027623, A037234, A037291, A037289, A038538, A186116.

max = 100; A000010 = EulerPhi[ Range[2*max]] // Union // Select[#, # <= max &] &; A181062 = Select[ Range[max], Length[ FactorInteger[#]] == 1 &] - 1; FixedPoint[ Select[ Outer[ Times, #, # ] // Flatten // Union, # <= max &] &, Union[A000010, A181062] ] (* Jean-François Alcover, Sep 10 2013 *)

list(lim)=my(P=1, q, v, u=List()); forprime(p=2, default(primelimit), if(eulerphi(P*=p)>=lim, q=p; break)); v=vecsort(vector(P/q*lim\eulerphi(P/q), k, eulerphi(k)), , 8); v=select(n->n<=lim, v); forprime(p=2, sqrtint(lim\1+1), P=p; while((P*=p) <= lim+1, listput(u, P-1))); v=vecsort(concat(v, Vec(u)), , 8); u=List([0]); while(#u, v=vecsort(concat(v, Vec(u)),,8); u=List(); for(i=3,#v, for(j=i,#v,P=v[i]*v[j]; if(P>lim,break); if(!vecsearch(v, P), listput(u, P))))); v \\ Charles R Greathouse IV, Jan 08 2013

Entry revised by N. J. A. Sloane, Jan 06 2013, Jan 08 2013

Definition clarified by Jianing Song, Dec 24 2021

A295584 Odd numbers that are not a product of Mersenne numbers (A000225).

Original entry on oeis.org

5, 11, 13, 17, 19, 23, 25, 29, 33, 35, 37, 39, 41, 43, 47, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 75, 77, 79, 83, 85, 87, 89, 91, 95, 97, 99, 101, 103, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 129, 131, 133, 137, 139, 141, 143, 145, 149, 151, 153

Offset: 1

N. J. A. Sloane, Dec 15 2017

nonn

Numbers m such that no commutative ring has m units.

Robert Israel, Table of n, a(n) for n = 1..10000
S. Chebolu and K. Lockridge, How Many Units Can a Commutative Ring Have?, Amer. Math. Monthly, 124 (2017), 960-965; arXiv, arXiv:1701.02341 [math.AC], 2017.

Odd numbers not in A282572; complement of A296241.

Cf. A000225.

N:= 1000: # to get all terms <= N
P:= {1}:
for k from 2 do
  m:= 2^k-1;
  if m > N then break fi;
  P:= map(p -> seq(p*m^j, j=0..floor(log[m](N/p))), P);
od:
sort(convert({seq(i,i=1..N,2)} minus P, list)); # Robert Israel, Dec 15 2017

nn == 1000;
P = {1};
For[k = 2, True, k++,
   m = 2^k - 1;
   If[m > nn, Break[]
];
P = (Function[p, Table[p m^j, {j, 0, Log[m, nn/p]}]] /@ P) // Flatten];
Range[1, nn, 2] ~Complement~ P (* Jean-François Alcover, Sep 18 2018, after Robert Israel *)

cp's OEIS Frontend

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

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A070932 Possible number of units in a finite (commutative or non-commutative) ring.

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A295584 Odd numbers that are not a product of Mersenne numbers (A000225).

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Maple

Mathematica