A346442 - OEIS

A346442

Decimal expansion, negated, of the location of the first occurrence of a negative minimum of the function P(x,m) = Sum_{k=1..m} prime(k)*x^(k-1), with m chosen such that P(x) has 2 real zeros, which occurs for m = 2436. The corresponding function value is A346443.

%I #20 Oct 17 2021 12:07:17

%S 9,9,6,6,4,3,5,1,3,7,3,7,6,7,2,9,2,7,8,2,0,2,8,5,8,8,8,4,3,3,3,7,6,8,

%T 1,3,4,5,1,3,3,8,9,5,1,4,1,5,1,7,7,5,8,0,6,0,6,2,9,2,8,8,4,9,3,9,5,7,

%U 5,9,2,8,9,2,1,0,9,8,5,5,3,6,5,2,9,3,8,4,6,6,8,2,3,7,3,3

%N Decimal expansion, negated, of the location of the first occurrence of a negative minimum of the function P(x,m) = Sum_{k=1..m} prime(k)*x^(k-1), with m chosen such that P(x) has 2 real zeros, which occurs for m = 2436. The corresponding function value is A346443.

%H W. Edwin Clark and Mark Shattuck, <a href="https://arxiv.org/abs/2107.05572">The Integer Sequence Transform a --> b where b_n is the Number of Real Roots of the Polynomial a_0 + a_1 x + a_2 x^2 + ... + a_n x^n</a>, arXiv:2107.05572 [math.CO], 2021.

%H Mathematics Stack Exchange, <a href="https://math.stackexchange.com/questions/4057417">An interesting observation on the roots of certain polynomials</a>, March 2021.

%H User "Linear Christmas" in Stack Exchange comments, <a href="https://i.stack.imgur.com/B588b.png">Graph of P(x,2436)</a>, Mar 18 2021.

%e -0.9966435137376729278202858884333768134513389514151775806...

%o (PARI) P(m,x)=sum(k=0,m,prime(k+1)*x^k);

%o d(m,y)=derivnum(x=y,P(m,x));

%o solve(x=-0.9967,-0.9966,d(2436,x))

%Y Cf. A346443, A346663.

%Y Positions of neighboring zeros: A346444, A346445.

%K nonn,cons

%O 0,1

%A _Hugo Pfoertner_, Jul 18 2021