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%I #15 Mar 08 2015 19:28:49
%S 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,
%T 43,233,29,811,1,941,101,1,1,1231,131,29,1,521,1,109,1,149,509,89,59,
%U 107,617,1,1,47,173,3067,47,1,3463,3599,89,431,4021,521,2161,2239,103,1,1
%N 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.
%C 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, ...).
%C a(58) = 3599 = 59*61 is not prime. - _T. D. Noe_, Nov 15 2006
%C Most terms are prime or 1.
%C 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, ...}.
%C Composite terms are listed in A141781 = {3599, 118477, 210589, 971573, 1164103, 1901959, 2446681, 3230069, ...}.
%C Note that all listed terms of A141781 are semiprime, for example: 3599 = 59*61, 118477 = 257*461, 210589 = 251*839, 971573 = 643*1511.
%C Conjecture: All composite terms are semiprime.
%H Alexander Adamchuk, Jul 04 2008, <a href="/A120292/b120292.txt">Table of n, a(n) for n = 1..282</a>
%F a(n) = Abs[numerator[Det[DiagonalMatrix[Table[Prime[i]/(Prime[i]+1)-1,{i,1,n}]]+1]]].
%F a(n) = Numerator[Abs[(-1)^n*(1 - Sum[Prime[k] + 1,{k, 1, n}])/Product[Prime[k] + 1, {k, 1, n}] ]],{n,1,282}]
%t Abs[Numerator[Table[Det[DiagonalMatrix[Table[Prime[i]/(Prime[i]+1)-1,{i,1,n}]]+1],{n,1,60}]]]
%t Table[Numerator[Abs[(1 - Sum[Prime[k] + 1,{k, 1, n}])/Product[Prime[k] + 1, {k, 1, n}] ]],{n,1,282}]
%o (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
%Y Cf. A024528, A118679, A125716, A141780, A141779, A141781.
%K frac,nonn
%O 1,1
%A _Alexander Adamchuk_, Jul 08 2006, Jul 04 2008
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