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A217788
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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.
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14
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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
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Edited and added additional information by Zhi-Wei Sun, Mar 31 2013
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STATUS
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approved
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