A062270 Numerators in partial products of the twin prime constant.

3, 45, 175, 693, 11011, 2807805, 302307005, 402243205, 714186915, 42803602439, 11086133031701, 5908908905896633, 1488200914442251997, 3041106216468949733, 16213234917387714257, 21611220383343195817

Offset: 2

Author

Frank Ellermann, Jun 16 2001

Keywords

• easy
• nonn
• frac

Comments

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

Examples

			a(4) = 175 = 3*1*5*3*7*5 / gcd(3*1*5*3*7*5, 2*2*4*4*6*6).
		

References

• Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 84-94.
• G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, ch. 22.20

Links

• Steven R. Finch, Hardy-Littlewood Constants [Broken link]
• Steven R. Finch, Hardy-Littlewood Constants [From the Wayback machine]

Crossrefs

Cf. A062271 (denominators), A005597 (decimal expansion).

Programs

Mathematica

Numerator[Abs[Table[ Det[ DiagonalMatrix[ Table[ 1/(Prime[i]-1)^2 - 1, {i, 1, n} ] ] + 1 ], {n, 2, 20} ]]] (* Alexander Adamchuk, Jun 02 2006 *)

PARI

a(n) = numerator(prod(k=2, n, 1-1/(prime(k)-1)^2)); \\ Michel Marcus, May 31 2022

Formula

a(n) = a(n-1)*(prime(n)*(prime(n)-2)) / gcd(a(n-1)*prime(n)*(prime(n)-2), A062271(n)) for n > 2.

Extensions

Typo in link corrected by Martin Griffiths, Apr 03 2009