A381604 Least number k such that A381596(k) = n.

1, 2, 5, 10, 17, 116, 501, 512

Offset: 0

Author

Michel Lagneau, Mar 01 2025

Keywords

• nonn
• hard
• more

Comments

Least number k such that the number of real zeros of the polynomial P(k,z) = Sum_{i=1..k} A001223(i)*z^(i-1) is equal to n, where A001223(i) = differences between consecutive primes.

Examples

			a(1) = 2 because P(2,z) = Sum_{i=1..2} A001223(i)*z^(i-1) = 1 + 2*z = 0 for z = -1/2.
a(2) = 5 because P(5,z) = Sum_{i=1..5} A001223(i)*z^(i-1) = 1 + 2*z + 2*z^2 + 4*z^3 + 2*z^4 = 0 for z = -1.6499348..., -0.5606729...
		

Crossrefs

Cf. A001223, A381596.

Programs

Maple

with(numtheory):
for n from 0 to 20 do:
ii:=0:
  for k from 1 to 10^3 while(ii=0) do :
    P:=add((ithprime(i+1)-ithprime(i))*x^(i-1),i=1..k):
    y:=fsolve(P,x,real):z:=evalf({%}):y:=nops(z):
     if y=n
     then
      ii:=1:printf (`%d %d \n`,n,k):
       else
     fi:
  od:
od: