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

Author

Cino Hilliard, May 08 2003

Keywords

• sign

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

Examples

			For n=4, we fit a cubic through the 4 points (1,2),(2,3),(3,5),(4,7) to obtain a(4) = 3.
		

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.

Crossrefs

Cf. A007442, A140119.

Programs

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}]

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 */

Formula

a(n) = sum{k=1, .., n} (-1)^(k+1) A007442(k)

Extensions

Edited by T. D. Noe, May 08 2003