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 A318189 a(n) is the least prime q such that P_n(q) = prime^n(q) + prime^(n-1)(q) + ... + prime^2(q) + prime(q) + q + (1-(-1)^n)/2 is prime, or -1 if no such prime exists. 0
 2, 5, 3, 13, 5, 3, 29, 3, 13, 13, 3, 5, 31, 7, 17 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS Definition: prime^0(q) = q; prime^r(q) = prime(prime^(r-1)(q)); r >= 1. P_n(q) can be thought of as a "primeth polynomial" of degree n, the sum of terms of ascending orders of primeness, such that the result is always an odd (possibly prime) number. Because of the rapid growth of the primeth function only the first few terms of this sequence have been found so far. Note: If q exists, a(n) > 2 for n >= 1 because P_n(2) is composite. LINKS N. Fernandez, An order of primeness, F(p), 1999. EXAMPLE For q = 2,3 P_1(q) = 6,9 respectively, but P_1(5) = 11 + 5 + 1 = 17; so a(1)=5. For q = 2, P_2(2) = 5 + 3 + 2 = 10, but P_2(3) = 11 + 5 + 3 = 19, so a(2) = 3. MAPLE P:=proc(n, q) add((ithprime@@j)(q), j=0..n)+(1-(-1)^n)/2; end: a:=proc(n) local q, i; for i from 1 do q:=ithprime(i): if isprime(P(n, q)) then return q; fi; od: end:  # W. Edwin Clark, Aug 20 2018 PROG (Python) from __future__ import division from sympy import isprime, prime, nextprime A318189_list, nmax, plist = [], 8, [[2]] for n in range(nmax):     r = (1-(-1)**n)//2     for x in plist:         if isprime(sum(x) + r):             A318189_list.append(x[0])             break     else:         p = plist[-1][0]         while True:             p = nextprime(p)             x = [p]             for i in range(n):                 x.append(prime(x[-1]))             plist.append(x)             if isprime(sum(x)+r):                 A318189_list.append(x[0])                 break     if n < nmax-1:         for x in plist:             x.append(prime(x[-1])) # Chai Wah Wu, Aug 20 2018 CROSSREFS Cf. A000720, A000040, A006450, A007097, A038580. Sequence in context: A124937 A279342 A169852 * A176914 A194010 A229609 Adjacent sequences:  A318186 A318187 A318188 * A318190 A318191 A318192 KEYWORD nonn,more AUTHOR David James Sycamore, Aug 19 2018 EXTENSIONS Terms a(0) and a(10) to a(14) were found by Edwin Clark, Hans Havermann, and Chai Wah Wu (Sequence Fans Mailing List, August, 2018). STATUS approved

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Last modified July 23 20:56 EDT 2019. Contains 325265 sequences. (Running on oeis4.)