A352296 Smallest number that can be expressed as the sum of two primes in exactly n ways or -1 if no such number exists.

1, 4, 10, 22, 34, 48, 60, 78, 84, 90, 114, 144, 120, 168, 180, 234, 246, 288, 240, 210, 324, 300, 360, 474, 330, 528, 576, 390, 462, 480, 420, 570, 510, 672, 792, 756, 876, 714, 798, 690, 1038, 630, 1008, 930, 780, 960, 870, 924, 900, 1134, 1434, 840, 990, 1302

Offset: 0

Author

Chai Wah Wu, Mar 11 2022

Keywords

• nonn

Comments

Conjecture: a(n) != -1 for all n.

If n > 0 and a(n) != -1, then a(n) is even.

a(0) = A014092(1)

a(1) = A067187(1)

a(2) = A067188(1)

a(3) = A067189(1)

a(4) = A067190(1)

a(5) = A067191(1)

a(6) = A066722(1)

a(7) = A352229(1)

a(8) = A352230(1)

a(9) = A352231(1)

a(10) = A352233(1)

Crossrefs

Cf. A014092, A067187, A067188, A067189, A067190, A067191, A066722, A352229, A352230, A352231, A352233.

Essentially the same as A023036 and A001172.

Programs

Mathematica

f[n_] := Count[IntegerPartitions[n, {2}], ?(And @@ PrimeQ[#] &)]; seq[max] :=  Module[{s = Table[0, {max}], n = 1, c = 0, k}, While[c < max, k = f[n]; If[k < max && s[[k + 1]] == 0, c++; s[[k + 1]] = n]; n++]; s]; seq[50] (* Amiram Eldar, Mar 11 2022 *)

Python

from itertools import count
from sympy import nextprime
def A352296(n):
    if n == 0:
        return 1
    pset, plist, pmax = {2}, [2], 4
    for m in count(2):
        if m > pmax:
            plist.append(nextprime(plist[-1]))
            pset.add(plist[-1])
            pmax = plist[-1]+2
        c = 0
        for p in plist:
            if 2*p > m:
                break
            if m - p in pset:
                c += 1
        if c == n:
            return m