A197625 Smallest set S of positive integers with following properties: (i) 1 is in S; (ii) if n,m are in S, so is n*m; (iii) if n,m are in S, so is n*m + m + n.

1, 3, 7, 9, 15, 19, 21, 27, 31, 39, 43, 45, 49, 55, 57, 63, 79, 81, 87, 91, 93, 99, 105, 111, 115, 117, 127, 129, 133, 135, 147, 159, 163, 165, 171, 175, 183, 187, 189, 199, 211, 217, 219, 223, 225, 231, 235, 237, 243, 255, 259, 261, 267, 271, 273, 279, 285

Offset

1,2

Comments

The set S in Bruck is slightly different because it does not include 1 and it includes a property "if n is in S, so is 2n + 1" which is a special case of the last property (iii) where m=1 since we allow 1 in S.

References

R. H. Bruck, What is a loop?, pp. 59-99 in A. A. Albert, ed., Studies in Modern Algebra, Vol. 2, Mathematical Association of America, 1963, see p. 67.

Links

Table of n, a(n) for n=1..57.

Prog

(PARI) {isA(n) = if( n<2, return( n==1 )); if( n%2 && isA( n\2 ), return( 1 )); fordiv( n, d, if( d*d > n, break); if( d==1, next); if( isA( d ) && isA( n/d ), return( 1 ))); fordiv( n+1, d, if( d*d > n, break); if( d==1, next); if( isA( d-1 ) && isA( (n+1)/d - 1 ), return( 1 )))}; {a(n) = local( cnt, m ); if( n<1, return( 0 )); while( cnt<n, if( isA( m++ ), cnt++ )); m}

Crossrefs

Sequence in context: A328012 A292911 A119681 * A190811 A104177 A099204

Adjacent sequences: A197622 A197623 A197624 * A197626 A197627 A197628

Keyword

nonn

Author

Michael Somos, Oct 16 2011

Status

approved