A070932 -id:A070932 - cp's OEIS frontend

A296241 Finite number of units in a commutative ring; nonnegative even numbers together with products of Mersenne numbers.

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, 34, 36, 38, 40, 42, 44, 45, 46, 48, 49, 50, 52, 54, 56, 58, 60, 62, 63, 64, 66, 68, 70, 72, 74, 76, 78, 80, 81, 82, 84, 86, 88, 90, 92, 93, 94, 96, 98, 100, 102, 104, 105, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126

Offset: 1

Jonathan Sondow, Dec 14 2017

nonn

Zero together with orders of finite abelian groups that appear as the group of units in a commutative ring (Chebolu and Lockridge).
Equals A005843 union A282572.
Also the possible number of units in a (commutative or non-commutative) ring, since every odd number that is the number of units of a ring must be in this sequence (Ditor's theorem, stated in the S. Chebolu and K. Lockridge link). - Jianing Song, Dec 24 2021

			The even integers {0, +-2, +-4, ...} form a commutative ring with no (multiplicative) units, so a(1) = 0.

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.

Cf. A005843, A282572.

A070932 is closely related.

A221178 Union of (prime powers minus 1) and values of Euler totient function.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 10, 12, 15, 16, 18, 20, 22, 24, 26, 28, 30, 31, 32, 36, 40, 42, 44, 46, 48, 52, 54, 56, 58, 60, 63, 64, 66, 70, 72, 78, 80, 82, 84, 88, 92, 96, 100, 102, 104, 106, 108, 110, 112, 116, 120, 124, 126, 127, 128, 130, 132, 136, 138, 140, 144, 148, 150, 156, 160, 162, 164, 166, 168, 172, 176

Offset: 1

Jean-François Alcover, Jan 06 2013

nonn

Cf. A000010, A002202, A000961, A181062, A070932 (multiplicative closure).

max = 200;
selNu = Select[Range[max], PrimeNu[#] == 1&]-1;
phiQ[m_] := Select[Range[m+1, 2*m*Product[1/(1-1/(k*Log[k])), {k, 2, DivisorSigma[0, m]}]], EulerPhi[#] == m&, 1] != {};
selPhi = Select[Range[max], phiQ];
Join[{0}, Union[selNu, selPhi]]

list(lim)=my(P=1, q, v, u=List([0])); 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))); vecsort(concat(v, Vec(u)),,8) \\ Charles R Greathouse IV, Jan 08 2013

Union of A181062 and A002202.

Edited by N. J. A. Sloane, Jan 06 2013

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A296241 Finite number of units in a commutative ring; nonnegative even numbers together with products of Mersenne numbers.

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A221178 Union of (prime powers minus 1) and values of Euler totient function.

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