A370090 Numbers that can be expressed in exactly one way as the unordered sum of two distinct primes.

5, 7, 8, 9, 10, 12, 13, 14, 15, 19, 21, 25, 31, 33, 38, 39, 43, 45, 49, 55, 61, 63, 69, 73, 75, 81, 85, 91, 99, 103, 105, 109, 111, 115, 129, 133, 139, 141, 151, 153, 159, 165, 169, 175, 181, 183, 193, 195, 199, 201, 213, 225, 229, 231, 235, 241, 243, 253, 259, 265

Offset

1,1

Comments

Apparently, a number that is the predecessor or successor of a prime number does not have a sum as defined here, except for a finite number of primes, which may be {7, 11, 13, 37}. - Peter Luschny, Feb 16 2024

Links

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Index entries for sequences related to Goldbach conjecture

Index entries for sequences related to partitions

Example

5 = 2+3; 7 = 2+5; 8 = 3+5; 9 = 2+7; 10 = 3+7 (10 = 5+5 is not considered).

Maple

select(n -> A117929(n) = 1, [seq(1..265)]);  # Peter Luschny, Feb 16 2024

Mathematica

tdpQ[{a_, b_}]:=AllTrue[{a, b}, PrimeQ]&&a!=b; Select[Range[300], Count[IntegerPartitions[#, {2}], _?tdpQ]==1&] (* Harvey P. Dale, Dec 30 2024 *)

Prog

(Python)
from sympy import sieve
from collections import Counter
from itertools import combinations
def aupton(max):
    sieve.extend(max)
    a = Counter(c[0]+c[1] for c in combinations(sieve._list, 2))
    return [n for n in range(1, max+1) if a[n] == 1]
print(aupton(265)) # Michael S. Branicky, Feb 16 2024

Crossrefs

Cf. A117929, A048974, A065091, A067187 (not necessarily distinct).

If we change 1 way (this sequence) we get A077914 (2 ways), A077969 (3 ways), A078299 (4 ways), A080854 (5 ways), and A080862 (6 ways).

Sequence in context: A243624 A038609 A078892 * A164374 A072281 A242273

Adjacent sequences: A370087 A370088 A370089 * A370091 A370092 A370093

Keyword

nonn

Author

Wesley Ivan Hurt, Feb 11 2024

Status

approved