A077914 -id:A077914 - cp's OEIS frontend

A117929 Number of partitions of n into 2 distinct primes.

0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 0, 2, 1, 2, 1, 2, 0, 3, 1, 2, 0, 2, 0, 3, 1, 2, 1, 3, 0, 4, 0, 1, 1, 3, 0, 4, 1, 3, 1, 3, 0, 5, 1, 4, 0, 3, 0, 5, 1, 3, 0, 3, 0, 6, 1, 2, 1, 5, 0, 6, 0, 2, 1, 5, 0, 6, 1, 4, 1, 5, 0, 7, 0, 4, 1, 4, 0, 8, 1, 4, 0, 4, 0, 9, 1, 4, 0, 4, 0, 7, 0, 3, 1, 6, 0, 8, 1, 5, 1

Offset: 1

Emeric Deutsch, Apr 03 2006

Number of distinct rectangles with prime length and width such that L + W = n, W < L. For example, a(16) = 2; the two rectangles are 3 X 13 and 5 X 11. - Wesley Ivan Hurt, Oct 29 2017

			a(24) = 3 because we have [19,5], [17,7] and [13,11].

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A010051, A045917, A061358, A073610, A166081 (positions of 0), A077914 (positions of 2), A080862 (positions of 6).

Column k=2 of A219180. - Alois P. Heinz, Nov 13 2012

g:=sum(sum(x^(ithprime(i)+ithprime(j)),i=1..j-1),j=1..35): gser:=series(g,x=0,130): seq(coeff(gser,x,n),n=1..125);
# alternative
A117929 := proc(n)
    local a,i,p ;
    a := 0 ;
    p := 2 ;
    for i from 1 do
        if 2*p >= n then
            return a;
        end if;
        if isprime(n-p) then
            a := a+1 ;
        end if;
        p := nextprime(p) ;
    end do:
end proc:
seq(A117929(n),n=1..80) ; # R. J. Mathar, Oct 01 2021

l = {}; For[n = 1, n <= 1000, n++, c = 0; For[k = 1, Prime[k] < n/2, k++, If[PrimeQ[n - Prime[k]], c = c + 1] ]; AppendTo[l, c] ] l (* Jake Foster, Oct 27 2008 *)
Table[Count[IntegerPartitions[n,{2}],?(AllTrue[#,PrimeQ]&&#[[1]]!= #[[2]] &)],{n,120}] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale, Jul 26 2020 *)

a(n)=my(s);forprime(p=2,(n-1)\2,s+=isprime(n-p));s \\ Charles R Greathouse IV, Feb 26 2014

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 [a[n] for n in range(1, max+1)]
print(aupton(105)) # Michael S. Branicky, Feb 16 2024

G.f.: Sum_{j>0} Sum_{i=1..j-1} x^(p(i)+p(j)), where p(k) is the k-th prime.
G.f.: A(x)^2/2 - A(x^2)/2 where A(x) = Sum_{p in primes} x^p. - Geoffrey Critzer, Nov 21 2012
a(n) = [x^n*y^2] Product_{i>=1} (1+x^prime(i)*y). - Alois P. Heinz, Nov 22 2012
a(n) = Sum_{i=2..floor((n-1)/2)} A010051(i) * A010051(n-i). - Wesley Ivan Hurt, Oct 29 2017

A080862 Numbers which can be expressed as the sum of two distinct primes in exactly six ways.

Original entry on oeis.org

60, 66, 72, 100, 110, 116, 172, 178, 182, 194, 206, 212, 218, 226, 248, 278, 326, 332, 398

Offset: 1

Shyam Sunder Gupta, Mar 29 2003

nonn

Apparently there are no further terms beyond 398. - R. J. Mathar, Oct 01 2021

			66 is a term as 66 = 37 + 29 = 43 + 23 = 47 + 17 = 53 + 13 = 59 + 7 = 61 + 5 are only the six ways to express 66 as a sum of two distinct primes.

Cf. A077914 (2 ways), A077969 (3 ways), A078299 (4 ways), A080854 (5 ways).

tdpQ[n_]:=Count[IntegerPartitions[n,{2}],?(AllTrue[#,PrimeQ]&&Length[Union[#]]==2&)]==6; Select[Range[400],tdpQ] (* _Harvey P. Dale, Aug 17 2025 *)

{j: A117929(j) = 6}. - R. J. Mathar, Oct 01 2021

A077969 Numbers which can be expressed as the sum of two distinct primes in exactly three ways.

Original entry on oeis.org

24, 30, 34, 40, 44, 46, 52, 56, 58, 98, 122, 128

Offset: 1

Shyam Sunder Gupta, Mar 29 2003

nonn

			30 is a term as 30 = 23+7 = 19+11 = 17+13 are the only three ways to express 30 as a sum of three distinct primes.

Cf. A077914 (2 ways), A078299 (4 ways), A080854 (5 ways), A080862 (6 ways).

A078299 Numbers which can be expressed as the sum of two distinct primes in exactly four ways.

Original entry on oeis.org

36, 42, 50, 74, 80, 82, 86, 88, 92, 94, 152, 158

Offset: 1

Shyam Sunder Gupta, Mar 29 2003

nonn

			36 is a term as 36 = 31 + 5 = 29 + 7 = 23 + 13 = 19 + 17 are only the four ways to express 36 as a sum of two distinct primes.

Cf. A077914 (2 ways), A077969 (3 ways), A080854 (5 ways), A080862 (6 ways).

A080854 Numbers which can be expressed as the sum of two distinct primes in exactly five ways.

Original entry on oeis.org

48, 54, 64, 70, 76, 104, 106, 118, 124, 134, 136, 146, 148, 164, 166, 188

Offset: 1

Shyam Sunder Gupta, Mar 29 2003

nonn

Apparently there are no further terms beyond 188. - R. J. Mathar, Oct 01 2021

			54 is a term as 54 = 31 + 23 = 37 + 17 = 41 + 13 = 43 + 11 = 47 + 7 are only the five ways to express 54 as a sum of two distinct primes.

Cf. A077914 (2 ways), A077969 (3 ways), A078299 (4 ways), A080862 (6 ways).

Select[Range[200],Count[IntegerPartitions[#,{2}],?(AllTrue[#,PrimeQ]&&#[[1]]!=#[[2]]&)]==5&] (* _Harvey P. Dale, Jul 14 2025 *)

{j: A117929(j) = 5} . - R. J. Mathar, Oct 01 2021

A344989 Smallest number whose number of partitions into n distinct primes is n, or zero if there are no such partitions.

Original entry on oeis.org

2, 16, 26, 33, 55, 59, 0, 0, 124, 159, 233, 227, 276, 0, 372, 480, 0, 0, 0, 752, 0, 920, 0, 1011, 0, 1211, 1425, 0, 0, 0, 0, 0, 2050, 2336, 2495, 0, 0, 0, 0, 3340, 0, 3712, 0, 0, 4303, 0, 0, 0, 0, 5195, 0, 5669, 0, 6163, 6673, 0, 0, 0, 7504, 0, 0, 8670, 0, 9304, 9623, 0, 0, 0, 10638, 10981, 0, 12062, 0

Offset: 1

Metin Sariyar, Jun 04 2021

From David A. Corneth, Aug 21 2025: (Start)
How to prove a 0? I used the heuristic:
a(n) = 0 if 2*n consecutive integers can be written in strictly more than n ways as a sum of n distinct primes and up to that point no positive integer has exactly n such ways.
What other rules where used? (End)

			a(2) = 16 because 16 is the smallest number whose number of partitions into 2 distinct primes is 2; 16 = 3+13 = 5+11.

Chris K. Caldwell and G. L. Honaker, Jr., Prime Curios! 233

Cf. A364692 asks for the largest number with the same properties.

Cf. A000586, A077914, A117929, A125688, A219180, A219198, A219199, A219200, A219201, A219202, A219203, A219204.

a(12)-a(20) from Alois P. Heinz, Jun 04 2021

More terms from David A. Corneth, Aug 21 2025

A352305 a(n) is the (conjectured) largest even number that can be expressed as the sum of two distinct primes in exactly n ways.

Original entry on oeis.org

6, 38, 68, 128, 158, 188, 398, 362, 458, 542, 632, 692, 602, 992, 808, 908, 1112, 1238, 1412, 1418, 1718, 1544, 1574, 1622, 1682, 2048, 2252, 2018, 2672, 2042, 2558, 2936, 2504, 2978, 2966, 3092, 3218, 3242, 3272, 3506, 3632, 3754, 4022, 4058, 4052, 4412, 4448, 4478

Offset: 0

Ilya Gutkovskiy, Mar 11 2022

nonn

			a(5) = 188 because 188 = 7 + 181 = 31 + 157 = 37 + 151 = 61 + 127 = 79 + 109 and it is conjectured that 188 is the last term of A080854.

Cf. A000954, A038609, A061357, A077914, A077969, A078299, A080854, A080862, A087747, A117929, A212613.

More terms from Hugo Pfoertner, Dec 18 2024

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

Original entry on oeis.org

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

Wesley Ivan Hurt, Feb 11 2024

nonn

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

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

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

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).

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

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

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

A140955 Even integers that do not have at least two partitions into 2 distinct primes.

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 12, 14, 38

Offset: 1

Gil Broussard, Jul 25 2008

If A056636(3) = 128 (as is conjectured), then 38 is the last term in the sequence. - Charles R Greathouse IV, Sep 07 2022

			8 is a term because 3+5 is the only sum of primes = 8.
16 is not in the sequence because 16 = 3+13 and 5+11.
The only ways to express 10 as a sum of two unordered primes are 3+7 and 5+5. In one of the sums the primes are distinct. Thus, 10 is in this sequence. - _Tanya Khovanova_, Sep 07 2022

Cf. A056636, A061357, A077914, A117929.

Select[Range[0,100,2],Length[Select[Union/@IntegerPartitions[#,{2}],AllTrue[#,PrimeQ]&&Length[#]==2&]]<2&] (* James C. McMahon, Jul 15 2025 *)

is(n)=if(n%2, return(0)); my(t); forprime(p=3, n\2-1, if(isprime(n-p) && t++>1, return(0))); 1 \\ Charles R Greathouse IV, Sep 07 2022

Offset changed to 1 by Alois P. Heinz, Sep 07 2022

A352596 Conjecturally the number of positive even integers that can be expressed as the sum of two distinct primes in exactly n ways.

Original entry on oeis.org

3, 5, 9, 12, 12, 16, 19, 13, 24, 19, 21, 25, 15, 29, 28, 16, 31, 22, 34, 32, 20, 29, 26, 24, 28, 36, 34, 35, 37, 22, 29, 37, 36, 34, 39, 32, 39, 35, 28, 31, 28

Offset: 0

Ilya Gutkovskiy, Mar 22 2022

Cf. A000974, A038609, A061357, A077914, A077969, A078299, A080854, A080862, A117929, A229492, A352305.

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A117929 Number of partitions of n into 2 distinct primes.

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A080862 Numbers which can be expressed as the sum of two distinct primes in exactly six ways.

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A077969 Numbers which can be expressed as the sum of two distinct primes in exactly three ways.

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A078299 Numbers which can be expressed as the sum of two distinct primes in exactly four ways.

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A080854 Numbers which can be expressed as the sum of two distinct primes in exactly five ways.

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A344989 Smallest number whose number of partitions into n distinct primes is n, or zero if there are no such partitions.

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A352305 a(n) is the (conjectured) largest even number that can be expressed as the sum of two distinct primes in exactly n ways.

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

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Maple

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Python

A140955 Even integers that do not have at least two partitions into 2 distinct primes.

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