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 A082594 Constant term when a polynomial of degree n-1 is fitted to the first n primes. 4
 2, 1, 2, 3, 6, 15, 38, 91, 206, 443, 900, 1701, 2914, 4303, 4748, 1081, -14000, -55335, -150394, -346163, -716966, -1369429, -2432788, -4002993, -5964748, -7525017, -6123026, 4900093, 40900520, 134308945, 348584680, 798958751, 1678213244, 3277458981, 5972923998, 10110994307 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS The polynomial is to pass through the points (k, prime(k)), k=1..n. 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. 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. REFERENCES J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 80 LINKS T. D. Noe, Table of n, a(n) for n=1..1000 Author?, Sicurvqf T. D. Noe, Plot of A082594 FORMULA a(n) = sum{k=1, .., n} (-1)^(k+1) A007442(k) EXAMPLE For n=4, we fit a cubic through the 4 points (1,2),(2,3),(3,5),(4,7) to obtain a(4) = 3. MATHEMATICA Table[Coefficient[Expand[InterpolatingPolynomial[Prime[Range[n]], x]], x, 0], {n, 50}] 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}] CROSSREFS Cf. A007442. Sequence in context: A001371 A001037 A122086 * A051850 A077013 A086880 Adjacent sequences:  A082591 A082592 A082593 * A082595 A082596 A082597 KEYWORD sign AUTHOR Cino Hilliard (hillcino368(AT)gmail.com), May 08 2003 EXTENSIONS Edited by T. D. Noe, May 08 2003 STATUS approved

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