A120292 Absolute value of numerator of determinant of n X n matrix with elements M[i,j] = prime(i)/(1+prime(i)) if i=j and 1 otherwise.

2, 1, 1, 5, 1, 23, 1, 1, 1, 23, 17, 13, 5, 1, 1, 1, 1, 37, 293, 47, 61, 29, 1, 29, 271, 593, 43, 233, 29, 811, 1, 941, 101, 1, 1, 1231, 131, 29, 1, 521, 1, 109, 1, 149, 509, 89, 59, 107, 617, 1, 1, 47, 173, 3067, 47, 1, 3463, 3599, 89, 431, 4021, 521, 2161, 2239, 103, 1, 1

Offset: 1

Author

Alexander Adamchuk, Jul 08 2006, Jul 04 2008

Keywords

• frac
• nonn

Comments

Some a(n) are equal to 1 (n = 2, 3, 5, 7, 8, 9, 14, 15, 16, 17, 23, 31, 34, 35, 39, 41, 43, 50, 51, 56, ...).

a(58) = 3599 = 59*61 is not prime. - T. D. Noe, Nov 15 2006

Most terms are prime or 1.

Numbers n such that a(n)>1 and is not prime are listed in A141779(n) = {58, 282, 367, 743, 808, 1015, 1141, 1299, 1962, 2109, 2179, 2397, 2501, ...}.

Composite terms are listed in A141781 = {3599, 118477, 210589, 971573, 1164103, 1901959, 2446681, 3230069, ...}.

Note that all listed terms of A141781 are semiprime, for example: 3599 = 59*61, 118477 = 257*461, 210589 = 251*839, 971573 = 643*1511.

Conjecture: All composite terms are semiprime.

Links

• Alexander Adamchuk, Jul 04 2008, Table of n, a(n) for n = 1..282

Crossrefs

Cf. A024528, A118679, A125716, A141780, A141779, A141781.

Programs

Mathematica

Abs[Numerator[Table[Det[DiagonalMatrix[Table[Prime[i]/(Prime[i]+1)-1,{i,1,n}]]+1],{n,1,60}]]]
Table[Numerator[Abs[(1 - Sum[Prime[k] + 1,{k, 1, n}])/Product[Prime[k] + 1, {k, 1, n}] ]],{n,1,282}]

PARI

a(n)=abs(numerator(matdet(matrix(n,n,i,j,if(i==j,prime(i)/(1+prime(i)),1))))) \\ Charles R Greathouse IV, Feb 07 2013