A019508 X^m=X rings without normal forms: integers m > 1 for which there exist a prime p and integers a,b > 0 such that both p^a-1 and p^b-1 divide m-1 but p^lcm(a,b)-1 does not divide m-1.

22, 43, 85, 94, 105, 106, 148, 169, 187, 209, 211, 218, 232, 274, 280, 295, 313, 316, 337, 358, 373, 382, 400, 417, 421, 435, 463, 466, 484, 521, 526, 547, 559, 589, 610, 625, 631, 652, 673, 715, 736, 745, 763, 778, 799, 833, 838, 841, 862, 869, 890, 904, 925, 931, 937

Offset

0,1

References

S. Burris and J. Lawrence, Term rewrite rules for finite fields, Internat. J. Algebra and Computation, 1 (1991), 353-369.

Links

Table of n, a(n) for n=0..54.

Example

22 is a member since 2^2-1 and 2^3-1 divide 22-1 but 2^6-1 does not.

Mathematica

Select[ Range[ 2, 1000 ], Function[ m, Module[ {k}, Length[ k=Flatten[ Select[ Map[ FactorInteger, 1+Divisors[ m-1 ] ], Length[ #1 ]==1&&#1[ [ 1, 2 ] ]>1& ], 1 ] ]>1&&Length[ Select[ Map[ Function[ p, {p, Last[ Transpose[ Select[ k, #1[ [ 1 ] ]==p& ] ] ]} ], Union[ First[ Transpose[ k ] ] ] ], Length[ #1[ [ 2 ] ] ]>1&&!MemberQ[ #1[ [ 2 ] ], LCM@@#1[ [ 2 ] ] ]& ] ]>0 ] ] ] (* Olivier Gérard *)

Crossrefs

Sequence in context: A118602 A215146 A086679 * A166058 A305609 A123799

Adjacent sequences: A019505 A019506 A019507 * A019509 A019510 A019511

Keyword

nonn

Author

Bill Dubuque (wgd(AT)martigny.ai.mit.edu)

Status

approved