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

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

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.

The dual sequence (in the sense of Sun, q.v.) of the primes. - Charles R Greathouse IV, Oct 03 2013

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

Zhi-Wei Sun, Combinatorial identities in dual sequences, European J. Combin. 24:6 (2003), pp. 709-718.

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}]

Prog

(PARI) dual(v:vec)=vector(#v, i, -sum(j=0, i-1, binomial(i-1, j)*(-1)^j*v[j+1]))
dual(concat(0, primes(100)))[2..101] \\ Charles R Greathouse IV, Oct 03 2013
(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 */

Crossrefs

Cf. A007442, A140119.

Sequence in context: A277633 A001037 A122086 * A051850 A077013 A086880

Adjacent sequences: A082591 A082592 A082593 * A082595 A082596 A082597

Keyword

sign

Author

Cino Hilliard, May 08 2003

Extensions

Edited by T. D. Noe, May 08 2003

Status

approved