A141779 Numbers k such that A120292(k) is composite.

58, 282, 367, 743, 808, 1015, 1141, 1299, 1962, 2109, 2179, 2397, 2501, 3704, 3825, 3912, 3932, 3935, 4016, 4049, 4247, 4327, 4598, 4915, 4977, 5210, 5266, 5396, 5420, 5512, 5562, 5773, 5981, 6031, 6249, 6616, 6984, 7117, 7121, 7304, 7338, 7424, 7653

Offset: 1

Author

Alexander Adamchuk, Jul 04 2008

Keywords

• nonn

Comments

Composite terms of A120292 are listed in A141781 = {3599, 118477, 210589, 971573, 1164103, 1901959, 2446681, 3230069, ...}.

Note that all listed terms correspond to semiprimes, for example: 3599 = 59*61, 118477 = 257*461, 210589 = 251*839, 971573 = 643*1511.

Conjecture: All composite terms of A120292 are semiprime.

Crossrefs

Cf. 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.

Cf. A125716 (k such that A120292(k) = 1).

Cf. A141780 (k such that A120292(k) is prime).

Cf. A141781 (terms of A120292 that are greater than 1 and are not prime; or A120292(A141779)).

Programs

Mathematica

Do[f=Numerator[Abs[(1 - Sum[Prime[k] + 1,{k, 1, n}])/Product[Prime[k] + 1, {k, 1, n}] ]];If[ !PrimeQ[f]&&!(f==1),Print[{n,f,FactorInteger[f]}]],{n,1,8212}]

PARI

for(n=1,100,t=abs(numerator(matdet(matrix(n,n,i,j,if(i==j, prime(i)/(1+prime(i)),1))))); if(t>3 && !isprime(t), print1(n", "))) \\ Charles R Greathouse IV, Feb 07 2013

Formula

A141781(n) = A120292( a(n) ).