A092894 Decimal expansion of Frederick Magata's constant.

3, 4, 0, 7, 0, 6, 9, 1, 6, 5, 6, 2, 7, 2, 5, 6, 1, 4, 2, 2, 1, 9, 4, 5, 8, 2, 6, 2, 8, 2, 7, 1, 8, 0, 6, 5, 3, 5, 5, 4, 0, 3, 4, 4, 3, 8, 0, 1, 5, 0, 3, 2, 1, 1, 6, 1, 9, 1, 0, 3, 3, 8, 2, 7, 5, 7, 2, 9, 6, 9, 9, 3, 8, 7, 0, 4, 1, 0, 3, 5, 1, 4, 3, 0, 0, 9, 9, 0, 0, 4, 0, 9, 3, 8, 9, 4, 7, 4, 1, 0, 8, 7, 8, 7, 1

Offset

1,1

Comments

Starting with the data (1,2), (2,3), (3,5), (4,7), (5,11), (6,13), ..., (n,p_n) where p_n is the n-th prime number, Frederick Magata (1998) used Newtonian interpolation to determine the coefficients b_k of a (n-1) degree polynomial fit b_0 + b_1*(x-1) + b_2*(x-1)*(x-2) + b_3*(x-1)*(x-2)*(x-3) + ... The sum of all the coefficients b_k, for arbitrarily large n, appears to converge to 3.407069...

Although it takes only 86 terms to secure the decimal representation above, Robert G. Wilson v took it out to 1000 terms.

References

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.5, p. 294.

Links

Table of n, a(n) for n=1..105.

S. R. Finch, Magata's constant.

Example

FM = 3.407069165627256142219458262827180653554034438015032116191033...

Mathematica

n := 100; P[y_] := P[y] = InterpolatingPolynomial[ Array[ Prime, n], x] /. x->y; fm = N[ Sum[ Level[ P[y], {2*k-1}] [[1]], {k, 1, n-1}] + Level[ P[y], {2*n-2}] [[1]], 111]; RealDigits[fm][[1]] (* Steven Finch *)

Crossrefs

Sequence in context: A110665 A063441 A319600 * A276563 A011338 A214024

Adjacent sequences: A092891 A092892 A092893 * A092895 A092896 A092897

Keyword

cons,nonn

Author

Robert G. Wilson v, Mar 10 2004

Status

approved