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%I #23 May 31 2022 11:23:36
%S 3,45,175,693,11011,2807805,302307005,402243205,714186915,42803602439,
%T 11086133031701,5908908905896633,1488200914442251997,
%U 3041106216468949733,16213234917387714257,21611220383343195817
%N Numerators in partial products of the twin prime constant.
%C For n>1, a(n) is the absolute value of the numerator of the determinant of the n X n matrix with elements M[i,j] = 1/(prime(i)-1)^2 for i=j and 1 otherwise. - _Alexander Adamchuk_, Jun 02 2006
%D Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 84-94.
%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, ch. 22.20
%H Steven R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/constant/hrdyltl/hrdyltl.html">Hardy-Littlewood Constants</a> [Broken link]
%H Steven R. Finch, <a href="http://web.archive.org/web/20010614100031/http://www.mathsoft.com/asolve/constant/hrdyltl/hrdyltl.html">Hardy-Littlewood Constants</a> [From the Wayback machine]
%F a(n) = a(n-1)*(prime(n)*(prime(n)-2)) / gcd(a(n-1)*prime(n)*(prime(n)-2), A062271(n)) for n > 2.
%e a(4) = 175 = 3*1*5*3*7*5 / gcd(3*1*5*3*7*5, 2*2*4*4*6*6).
%t Numerator[Abs[Table[ Det[ DiagonalMatrix[ Table[ 1/(Prime[i]-1)^2 - 1, {i, 1, n} ] ] + 1 ], {n, 2, 20} ]]] (* _Alexander Adamchuk_, Jun 02 2006 *)
%o (PARI) a(n) = numerator(prod(k=2, n, 1-1/(prime(k)-1)^2)); \\ _Michel Marcus_, May 31 2022
%Y Cf. A062271 (denominators), A005597 (decimal expansion).
%K easy,nonn,frac
%O 2,1
%A _Frank Ellermann_, Jun 16 2001
%E Typo in link corrected by _Martin Griffiths_, Apr 03 2009