

A217788


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


14



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
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OFFSET

1,1


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^{nk} (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^(nk) 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.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..1000
ZhiWei Sun, Problems on irreducible polynomials, a message to Number Theory List, March 24, 2013.
ZhiWei Sun, Primes of the form 1+2*s+...+n*s^{n1}, a message to Number Theory List, March 24, 2013.


EXAMPLE

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.


MATHEMATICA

A[n_, x_]:=A[n, x]=Sum[Prime[k]*x^(nk), {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}]


CROSSREFS

Cf. A000040, A217785, A218465, A220072, A223934, A224197.
Sequence in context: A319875 A123722 A328733 * A273257 A249485 A254877
Adjacent sequences: A217785 A217786 A217787 * A217789 A217790 A217791


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Mar 25 2013


EXTENSIONS

Edited and added additional information by ZhiWei Sun, Mar 31 2013


STATUS

approved



