A335435 Triangle read by rows: T(n,k) is the least degree of a polynomial interpolating (i, prime(i)) for i from k to n.

0, 1, 0, 2, 1, 0, 3, 1, 1, 0, 4, 3, 2, 1, 0, 5, 4, 3, 2, 1, 0, 6, 5, 4, 3, 2, 1, 0, 7, 6, 5, 4, 3, 2, 1, 0, 8, 7, 6, 5, 4, 3, 2, 1, 0, 9, 8, 7, 6, 5, 4, 2, 2, 1, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9

Offset

1,4

Comments

The prime k-tuples conjecture implies that, for each k >= 2, each j from 1 to k-1 occurs infinitely often as T(n,n-k).

Links

Robert Israel, Table of n, a(n) for n = 1..10011 (rows 1 to 141, flattened)

MathOverflow, Polynomials of minimum degree that interpolate primes in intervals

Example

Triangle begins:
0
1 0
2 1 0
3 1 1 0
4 3 2 1 0
5 4 3 2 1 0
For n=10 and k=7, the points to interpolate are [7, 17], [8, 19], [9, 23], [10, 29].  These are interpolated by the polynomial y = x^2 - 13*x + 59 of degree 2, so T(10,7) = 2.

Maple

g:= proc(n, m) local L, d;
  L:= [seq(ithprime(i), i=m..n)];
  for d from 0 do
    L:= L[2..-1]-L[1..-2];
    if L = [] or convert(L, set) = {0} then return d fi
  od
end proc:
for n from 1 to 10 do
  seq(g(n, m), m=1..n)
od;

Crossrefs

Sequence in context: A176808 A327029 A167192 * A363048 A262114 A320780

Adjacent sequences: A335432 A335433 A335434 * A335436 A335437 A335438

Keyword

nonn,tabl

Author

Robert Israel, Jul 01 2020

Status

approved