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.

2, 5, 3, 13, 5, 3, 29, 3, 13, 13, 3, 5, 31, 7, 17

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

Table of n, a(n) for n=0..14.

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, Aug 22 2018).

Status

approved