Search: id:A082594
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%I A082594
%S A082594 2,1,2,3,6,15,38,91,206,443,900,1701,2914,4303,4748,1081,14000,55335,150394,
               346163,
%T A082594 716966,1369429,2432788,4002993,5964748,7525017,6123026,4900093,40900520,
%U A082594 134308945,348584680,798958751,1678213244,3277458981,5972923998,10110994307
%V A082594 2,1,2,3,6,15,38,91,206,443,900,1701,2914,4303,4748,1081,-14000,-55335,
               -150394,-346163,
%W A082594 -716966,-1369429,-2432788,-4002993,-5964748,-7525017,-6123026,4900093,
               40900520,
%X A082594 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.
%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="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%20F\
               itting&fc_a=0">Sicurvqf</a>
%H A082594 T. D. Noe, <a href="http://www.sspectra.com/math/A082594.gif">Plot of 
               A082594</a>
%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}]
%Y A082594 Cf. A007442.
%Y A082594 Sequence in context: A001371 A001037 A122086 this_sequence A051850 A077013 
               A086880
%Y A082594 Adjacent sequences: A082591 A082592 A082593 this_sequence A082595 A082596 
               A082597
%K A082594 sign
%O A082594 1,1
%A A082594 Cino Hilliard (hillcino368(AT)gmail.com), May 08 2003
%E A082594 Edited by T. D. Noe (noe(AT)sspectra.com), May 08 2003

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