A067188 -id:A067188 - cp's OEIS frontend

A352233 Numbers that can be expressed as the sum of two primes in exactly 10 ways.

114, 126, 162, 260, 290, 304, 316, 328, 344, 352, 358, 374, 382, 416, 542, 632

Offset: 1

Wesley Ivan Hurt, Mar 08 2022

All terms are even. Conjecture: 632 is the last term. Hardy and Littlewood conjectured a growth rate of the number of decompositions for large even numbers (see Conjecture A in page 32 of Hardy and Littlewood reference), implying this sequence is finite. - Chai Wah Wu, Mar 10 2022

			114 = 5+109 = 7+107 = 11+103 = 13+101 = 17+97 = 31+83 = 41+73 = 43+71 = 47+67 = 53+61.

G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes, Acta Mathematica, volume 44, pages 1-70 (1923).

Numbers that can be expressed as the sum of two primes in k ways for k=0..10: A014092 (k=0), A067187 (k=1), A067188 (k=2), A067189 (k=3), A067190 (k=4), A067191 (k=5), A066722 (k=6), A352229 (k=7), A352230 (k=8), A352231 (k=9), this sequence (k=10).

c[n_] := Count[IntegerPartitions[n, {2}], ?(And @@ PrimeQ[#] &)]; Select[Range[1000], c[#] == 10 &] (* _Amiram Eldar, Mar 08 2022 *)

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

Original entry on oeis.org

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

Chai Wah Wu, Mar 11 2022

nonn

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)

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

Essentially the same as A023036 and A001172.

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 *)

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

cp's OEIS Frontend

A352233 Numbers that can be expressed as the sum of two primes in exactly 10 ways.

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