A217788 Least integer s > p_n such that sum_{k=1}^n p_k*s^(n-k) (the number [p_1,...,p_n] in base s) is prime, where p_k denotes the k-th prime.

3, 4, 8, 9, 16, 15, 72, 37, 30, 54, 54, 54, 80, 91, 78, 204, 182, 110, 286, 183, 158, 231, 228, 105, 252, 189, 198, 119, 178, 252, 280, 152, 164, 423, 170, 185, 190, 249, 1006, 249, 678, 200, 254, 480, 216, 234, 322, 601, 264, 301, 260, 269, 244, 308, 280, 364, 612, 635, 310, 420

Offset: 1

Author

Zhi-Wei Sun, Mar 25 2013

Keywords

• nonn

Comments

Conjecture: For any integers n >= m > 0, there are infinitely many positive integers s > p_n such that the number sum_{k=m}^n p_k*s^{n-k} (i.e., [p_m,...,p_n] in base s) is prime; moreover the smallest such an integer s (denoted by s(m,n)) does not exceed (n+1)*(m+n+1).

Note that s(1,n) = a(n) and s(4,21) = 546 < (21+1)*(21+4+1) = 572.

A related conjecture of the author states that for each n=2,3,... the polynomial sum_{k=1}^n p_k*x^(n-k) is irreducible modulo some prime. See also the author's comments on A000040.

The conjecture can be further extended as follows: If a_1 < ... < a_n are distinct integers with a_n prime, then there are infinitely many integers b > a_n such that [a_1,a_2,...,a_n] in base b is prime.

For example, [2,3,...,210,211] in base 55272 and[17,19,27,34,38,41] in base 300 are both prime.

See A224197 for a more general conjecture.

Examples

			a(3)=8 since 2*8^2+3*8+5=157 is prime but 2*6^2+3*6+5=95 and 2*7^2+3*7+5=124 are not.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..1000
• Zhi-Wei Sun, Problems on irreducible polynomials, a message to Number Theory List, March 24, 2013.
• Zhi-Wei Sun, Primes of the form 1+2*s+...+n*s^{n-1}, a message to Number Theory List, March 24, 2013.

Crossrefs

Cf. A000040, A217785, A218465, A220072, A223934, A224197.

Programs

Mathematica

A[n_,x_]:=A[n,x]=Sum[Prime[k]*x^(n-k),{k,1,n}]; Do[Do[If[PrimeQ[A[n,s]]==True, Print[n," ",s]; Goto[aa]], {s,Prime[n]+1, (n+1)(n+2)}]; Print[n," ",counterexample]; Label[aa]; Continue, {n,1,100}]

Extensions

Edited and added additional information by Zhi-Wei Sun, Mar 31 2013