cp's OEIS Frontend

A082594 Constant term when a polynomial of degree n-1 is fitted to the first n primes.

%I A082594 #23 Dec 02 2020 16:43:27
%S A082594 2,1,2,3,6,15,38,91,206,443,900,1701,2914,4303,4748,1081,-14000,
%T A082594 -55335,-150394,-346163,-716966,-1369429,-2432788,-4002993,-5964748,
%U A082594 -7525017,-6123026,4900093,40900520,134308945,348584680,798958751,1678213244,3277458981,5972923998,10110994307
%N A082594 Constant term when a polynomial of degree n-1 is fitted to the first n primes.
%C A082594 The polynomial is to pass through the points (k, prime(k)), k=1..n.
%C A082594 The constant term is always an integer because it is the same as f(0), which can be computed from the difference table of the sequence of primes. See Conway and Guy. In fact, the interpolating polynomial is integral for all integer arguments.
%C A082594 A plot of the first 1000 terms shows that the sequence grows exponentially and changes signs occasionally. The Mathematica lines show two ways of computing the sequence. The second, which uses the difference table, is much faster.
%C A082594 The dual sequence (in the sense of Sun, q.v.) of the primes. - _Charles R Greathouse IV_, Oct 03 2013
%D A082594 J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 80
%H A082594 T. D. Noe, <a href="/A082594/b082594.txt">Table of n, a(n) for n = 1..1000</a>
%H A082594 Author?, <a href="http://groups.msn.com/BC2LCC/page.msnw?fc_p=%2FSicurv%20%2D%20Simul%20Equ%20and%20Curve%20Fitting&amp;fc_a=0">Sicurvqf</a>
%H A082594 T. D. Noe, <a href="http://www.sspectra.com/math/A082594.gif">Plot of A082594</a>
%H A082594 Zhi-Wei Sun, <a href="http://maths.nju.edu.cn/~zwsun/60e.pdf">Combinatorial identities in dual sequences</a>, European J. Combin. 24:6 (2003), pp. 709-718.
%F A082594 a(n) = sum{k=1, .., n} (-1)^(k+1) A007442(k)
%e A082594 For n=4, we fit a cubic through the 4 points (1,2),(2,3),(3,5),(4,7) to obtain a(4) = 3.
%t A082594 Table[Coefficient[Expand[InterpolatingPolynomial[Prime[Range[n]], x]], x, 0], {n, 50}]
%t A082594 Diff[lst_List] := Table[lst[[i+1]]-lst[[i]], {i, Length[lst]-1}]; n=50; dt=Table[{}, {n}]; dt[[1]]=Prime[Range[n]]; Do[dt[[i]]=Diff[dt[[i-1]]], {i, 2, n}]; Table[s=dt[[i, 1]]; Do[s=dt[[i-j, 1]]-s, {j, i-1}]; s, {i, n}]
%o A082594 (PARI) dual(v:vec)=vector(#v,i,-sum(j=0,i-1,binomial(i-1,j)*(-1)^j*v[j+1]))
%o A082594 dual(concat(0,primes(100)))[2..101] \\ _Charles R Greathouse IV_, Oct 03 2013
%o A082594 (PARI) {a(n) = sum(k=0, n-1, sum(i=0, k, binomial(k, i) * (-1)^i * prime(i+1)))}; /* _Michael Somos_, Dec 02 2020 */
%Y A082594 Cf. A007442, A140119.
%K A082594 sign
%O A082594 1,1
%A A082594 _Cino Hilliard_, May 08 2003
%E A082594 Edited by _T. D. Noe_, May 08 2003