A117929 -id:A117929 - cp's OEIS frontend

A219180 Number T(n,k) of partitions of n into k distinct prime parts; triangle T(n,k), n>=0, read by rows.

1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 2, 1, 0, 1, 0, 0, 1, 0, 0, 2, 2, 0, 1, 1, 1, 0, 0, 2, 2, 0, 0, 1, 2, 1, 0, 0, 2, 2, 0, 1, 0, 2, 2, 0, 0, 3, 2, 0, 0, 1, 2, 2, 0, 0, 2, 3, 1

Offset: 0

Alois P. Heinz, Nov 13 2012

T(n,k) is defined for all n>=0 and k>=0. The triangle contains only elements with 0 <= k <= A024936(n). T(n,k) = 0 for k > A024936(n). Three rows are empty because there are no partitions of n into distinct prime parts for n in {1,4,6}.

			T(0,0) = 1: [], the empty partition.
T(2,1) = 1: [2].
T(5,1) = 1: [5], T(5,2) = 1: [2,3].
T(16,2) = 2: [5,11], [3,13].
Triangle T(n,k) begins:
  1;
  ;
  0, 1;
  0, 1;
  ;
  0, 1, 1;
  ;
  0, 1, 1;
  0, 0, 1;
  0, 0, 1;
  0, 0, 1, 1;
  0, 1;
  0, 0, 1, 1;
  ...

Alois P. Heinz, Rows n = 0..1000, flattened

Columns k=0-10 give: A000007, A010051, A117929, A125688, A219198, A219199, A219200, A219201, A219202, A219203, A219204.
Row lengths are 1 + A024936(n).
Row sums give: A000586.
Last elements of rows give: A219181.
Row maxima give: A219182.
Least n with T(n,k) > 0 is A007504(k).
Cf. A000040, A117278, A219107.

b:= proc(n, i) option remember;
      `if`(n=0, [1], `if`(i<1, [], zip((x, y)->x+y, b(n, i-1),
       [0, `if`(ithprime(i)>n, [], b(n-ithprime(i), i-1))[]], 0)))
    end:
T:= proc(n) local l; l:= b(n, numtheory[pi](n));
       while nops(l)>0 and l[-1]=0 do l:= subsop(-1=NULL, l) od; l[]
    end:
seq(T(n), n=0..50);

nn=20;a=Table[Prime[n],{n,1,nn}];CoefficientList[Series[Product[1+y x^a[[i]],{i,1,nn}],{x,0,nn}],{x,y}]//Grid  (* Geoffrey Critzer, Nov 21 2012 *)
zip[f_, x_List, y_List, z_] := With[{m = Max[Length[x], Length[y]]}, f[PadRight[x, m, z], PadRight[y, m, z]]]; b[n_, i_] := b[n, i] = If[n == 0, {1}, If[i<1, {}, zip[Plus, b[n, i-1], Join[{0}, If[Prime[i] > n, {}, b[n-Prime[i], i-1]]], 0]]]; T[n_] := Module[{l}, l = b[n, PrimePi[n]]; While[Length[l]>0 && l[[-1]] == 0, l = ReplacePart[l, -1 -> Sequence[]]]; l]; Table[T[n], {n, 0, 50}] // Flatten (* Jean-François Alcover, Jan 29 2014, after Alois P. Heinz *)

T(n)={ Vec(prod(k=1, n, 1 + isprime(k)*y*x^k + O(x*x^n))) }
{ my(t=T(20)); for(n=1, #t, print(if(t[n]!=0, Vecrev(t[n]), []))) } \\ Andrew Howroyd, Dec 22 2017

G.f. of column k: Sum_{0

T(n,k) = [x^n*y^k] Product_{i>=1} (1+x^prime(i)*y).

A061357 Number of 0

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 2, 2, 3, 2, 3, 4, 1, 3, 4, 3, 3, 5, 4, 3, 5, 3, 3, 6, 2, 5, 6, 2, 5, 6, 4, 5, 7, 4, 4, 8, 4, 4, 9, 4, 4, 7, 3, 6, 8, 5, 5, 8, 6, 7, 10, 6, 5, 12, 3, 5, 10, 3, 7, 9, 5, 5, 8, 7, 7, 11, 5, 5, 12, 4, 8, 11, 4, 8, 10, 5, 5, 13, 9, 6, 11, 7, 6, 14, 6, 8, 13, 5, 8, 11, 6, 9

Offset: 1

Amarnath Murthy, Apr 28 2001

Number of prime pairs (p,q) with p < n < q and q-n = n-p.
The same as the number of ways n can be expressed as the mean of two distinct primes.
Conjecture: for n>=4 a(n)>0. - Benoit Cloitre, Apr 29 2003
Conjectures from Rick L. Shepherd, Jun 24 2003: (Start)
1) For each integer N>=1 there exists a positive integer m(N) such that for n>=m(N) a(n)>a(N). (After the first m(N)-1 terms, a(N) does not reappear). In particular, for N=1 (or 2 or 3), m(N)=4 and a(N)=0, giving Benoit Cloitre's conjecture. (cont.)
(cont.) Conjectures based upon observing a(1),...,a(10000):
m(4)=m(5)=m(6)=m(7)=m(19)=20 for a(4)=a(5)=a(6)=a(7)=a(19)=1,
m(8)=...(7 others)...=m(34)=35 for a(8)=...(7 others)...=a(34)=2,
m(12)=...(10 others)...=m(64)=65 for a(12)=...(10 others)...=a(64)=3,
m(18)=...(10 others)...=m(79)=80 for a(18)=...(10 others)...=a(79)=4,
m(24)=...(14 others)...=m(94)=95 for a(24)=...(14 others)...=a(94)=5,
m(30)=...(17 others)...=m(199)=200 for a(30)=...(17 others)...=a(199)=6, etc.
2) Each nonnegative integer appears at least once in the current sequence.
3) Stronger than 2): A001477 (nonnegative integers) is a subsequence of the current sequence. (Supporting evidence: I've observed that 0,1,2,...,175 is a subsequence of a(1),...,a(10000)).
(End)
a(n) is also the number of k such that 2*k+1=p and 2*(n-k-1)+1=q are both odd primes with p < q with p*q = n^2 - m^2. [Pierre CAMI, Sep 01 2008]
Also: Number of ways n^2 can be written as b^2+pq where 0a(n) = sum (A010051(2*n - p): p prime < n). [Reinhard Zumkeller, Oct 19 2011]
a(n) is also the number of partitions of 2*n into two distinct primes. See the first formula by T. D. Noe, and the Alois P. Heinz, Nov 14 2012, crossreference. - Wolfdieter Lang, May 13 2016
All 0Jamie Morken, Jun 02 2017
a(n) is the number of appearances of n in A143836. - Ya-Ping Lu, Mar 05 2023

			a(10)= 2: there are two such pairs (3,17) and (7,13), as 10 = (3+17)/2 = (7+13)/2.

Pierre CAMI, Table of n, a(n) for n = 1..60000

Cf. A071681 (subsequence for prime n only).
Cf. A092953.
Bisection of A117929 (even part). - Alois P. Heinz, Nov 14 2012

a061357 n = sum $
   zipWith (\u v -> a010051 u * a010051 v) [n+1..] $ reverse [1..n-1]
-- Reinhard Zumkeller, Nov 10 2012, Oct 19 2011

Table[Count[Range[n - 1], k_ /; And[PrimeQ[n - k], PrimeQ[n + k]]], {n, 98}] (* Michael De Vlieger, May 14 2016 *)

a(n)=my(s);forprime(p=2,n-1,s+=isprime(2*n-p));s \\ Charles R Greathouse IV, Mar 08 2013

from sympy import primerange, isprime
def A061357(n): return sum(1 for p in primerange(n) if isprime((n<<1)-p)) # Chai Wah Wu, Sep 03 2024

a(n) = A045917(n) - A010051(n). - T. D. Noe, May 08 2007
a(n) = sum(A010051(n-k)*A010051(n+k): 1 <= k < n). - Reinhard Zumkeller, Nov 10 2012
a(n) = sum_{i=2..n-1} A010051(i)*A010051(2n-i). [Wesley Ivan Hurt, Aug 18 2013]

More terms from Larry Reeves (larryr(AT)acm.org), May 15 2001

A077914 Sums of two distinct primes in exactly two ways.

Original entry on oeis.org

16, 18, 20, 22, 26, 28, 32, 62, 68

Offset: 1

Shyam Sunder Gupta, Mar 29 2003

Most likely no more terms. See A117929. - T. D. Noe, Mar 21 2012

Numbers k such that A117929(k) = 2. - Joerg Arndt, Jun 07 2021

			22 is a term as 22 = 19+3 = 17+5 are the only two ways to express 22 as a sum of two distinct primes.

G. L. Honaker, Jr., Prime Curio for 16, April 2000.

Cf. A077969 (3 ways), A078299 (4 ways), A080854 (5 ways), A080862 (6 ways).

A166081 Natural numbers that are not the sum of two distinct primes.

Original entry on oeis.org

1, 2, 3, 4, 6, 11, 17, 23, 27, 29, 35, 37, 41, 47, 51, 53, 57, 59, 65, 67, 71, 77, 79, 83, 87, 89, 93, 95, 97, 101, 107, 113, 117, 119, 121, 123, 125, 127, 131, 135, 137, 143, 145, 147, 149, 155, 157, 161, 163, 167, 171, 173, 177, 179, 185, 187, 189, 191, 197, 203

Offset: 1

Juri-Stepan Gerasimov, Oct 06 2009

nonn

All numbers that appear in A014092 are also in this sequence, by definition.
It seems that, for n > 6, the reverse is also true, however this is unproved. - Ely Golden, Dec 25 2016
All numbers that appear in this sequence but not A014092 must be even semiprimes with no other partitions into primes. - Ely Golden, Dec 25 2016

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from G. C. Greubel)

Cf. A117929, A006881, A038609 (complement), A014092, A066615.

Select[Range@ 204, Length@Select[Transpose@{#, Reverse@ # - 1} &@ Range[#] &@ #, Times @@ Boole@ Map[PrimeQ, #] == 1 && First@ # != Last@ # &] == 0 &] (* Michael De Vlieger, Apr 24 2016 *)
max = 1000;
ip = PrimePi[max];
A038609 = Table[Prime[i] + Prime[j], {i, ip}, {j, i + 1, ip}] // Flatten // Union // Select[#, # <= max&]&;
Complement[Range[max], A038609] (* Jean-François Alcover, Mar 24 2020 *)

{1} U A025584 U A109934. - R. J. Mathar, Oct 08 2009

A000027 \ A038609. - R. J. Mathar, Oct 14 2009

A341973 Number of partitions of n into 2 distinct primes (counting 1 as a prime).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 0, 2, 1, 2, 1, 2, 0, 3, 1, 3, 1, 2, 0, 4, 1, 2, 0, 2, 0, 4, 1, 3, 1, 3, 0, 4, 0, 2, 1, 3, 0, 5, 1, 4, 1, 3, 0, 6, 1, 4, 0, 3, 0, 6, 1, 3, 0, 3, 0, 7, 1, 3, 1, 5, 0, 6, 0, 3, 1, 5, 0, 7, 1, 5, 1, 5, 0, 7, 0, 5, 1, 4, 0, 9, 1, 4, 0, 4, 0, 10, 1, 4, 0, 4, 0, 7

Offset: 3

Ilya Gutkovskiy, Feb 24 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..10000

Cf. A008578, A036497, A117929, A154804, A341945, A341974, A341975, A341976, A341977.

b:= proc(n, i) option remember; series(`if`(n=0, 1,
     `if`(i<0, 0, (p-> `if`(p>n, 0, x*b(n-p, i-1)))(
     `if`(i=0, 1, ithprime(i)))+b(n, i-1))), x, 3)
    end:
a:= n-> coeff(b(n, numtheory[pi](n)), x, 2):
seq(a(n), n=3..96);  # Alois P. Heinz, Feb 24 2021

a[n_] := Select[IntegerPartitions[n, {2}, Join[{1},
    Prime[Range[PrimePi[n-1]]]]], #[[1]] != #[[2]]&] // Length;
a /@ Range[3, 100] (* Jean-François Alcover, Jul 13 2021 *)

a(n) = A117929(n) + A010051(n-1). - R. J. Mathar, Oct 01 2021

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

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

A087747 Least number expressible as a sum of two distinct primes in exactly n ways.

Original entry on oeis.org

5, 16, 24, 36, 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, 1080, 1230, 1518

Offset: 1

Lekraj Beedassy, Oct 02 2003

nonn

			a(2)=16 because we have 16=3+13=5+11, followed by 18=5+13=7+11; 20=3+17=7+13; 22=3+19=5+17; 26=3+23=7+19;...

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Cf. A117929.

Module[{nn=2000,ip},ip=Table[{n,Length[Select[IntegerPartitions[n,{2}], #[[1]]!=#[[2]]&&AllTrue[#,PrimeQ]&]]},{n,nn}];Table[SelectFirst[ip, #[[2]]==k&],{k,60}]][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 11 2020 *)

a(n) = min({m >= 1 : A117929(m) = n}). - Peter Munn, May 01 2023

More terms from John Dethridge (jcd(AT)ms.unimelb.edu.au), Jan 08 2004
More terms from Hugo Pfoertner, Sep 23 2004
Reinserted a(18), a(19) and a(20) by Paolo P. Lava, May 26 2014

A243485 Sum of all the products formed by multiplying the corresponding smaller and larger parts of the Goldbach partitions of n.

Original entry on oeis.org

0, 0, 0, 4, 6, 9, 10, 15, 14, 46, 0, 35, 22, 82, 26, 94, 0, 142, 34, 142, 38, 263, 0, 357, 46, 371, 0, 302, 0, 591, 58, 334, 62, 780, 0, 980, 0, 578, 74, 821, 0, 1340, 82, 785, 86, 1356, 0, 1987, 94, 1512, 0, 1353, 0, 2677, 106, 1421, 0, 2320, 0, 4242, 118

Offset: 1

Wesley Ivan Hurt, Jun 05 2014

a(n) is even for odd n.
If Goldbach's conjecture is true, a(n) > 0 for all even n > 2.
Sum of the areas of the distinct rectangles with prime length and width such that L + W = n, W <= L. For example, a(16) = 94; the two rectangles are 3 X 13 and 5 X 11, and the sum of their areas is 3*13 + 5*11 = 94. - Wesley Ivan Hurt, Oct 28 2017

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Goldbach Partition
Wikipedia, Goldbach's conjecture
Index entries for sequences related to Goldbach conjecture
Index entries for sequences related to partitions

Cf. A002372, A002375, A045917, A064911, A117929.

with(numtheory): A243485:=n->add(i*(n-i)*(pi(i)-pi(i-1))*(pi(n-i)-pi(n-i-1)), i=1..floor(n/2)): seq(A243485(n), n=1..100); # Wesley Ivan Hurt, Oct 29 2017

Table[Sum[i*(n - i)*Floor[2/PrimeOmega[i (n - i)]], {i, 2, n/2}], {n,
  50}]

a(n) = Sum_{i=2..n/2} i*(n-i) * A064911(i*(n-i)).

a(n) = Sum_{i=1..floor(n/2)} i * (n-i) * A010051(i) * A010051(n-i). - Wesley Ivan Hurt, Oct 29 2017

A347550 Number of partitions of n into at most 2 distinct prime parts.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Sep 08 2021

nonn

Cf. A000040, A000586, A117929, A219180, A223893.

a(n) = Sum_{k=0..2} A219180(n,k). - Alois P. Heinz, Sep 08 2021

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A219180 Number T(n,k) of partitions of n into k distinct prime parts; triangle T(n,k), n>=0, read by rows.

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A061357 Number of 0

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A077914 Sums of two distinct primes in exactly two ways.

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A166081 Natural numbers that are not the sum of two distinct primes.

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A341973 Number of partitions of n into 2 distinct primes (counting 1 as a prime).

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

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A087747 Least number expressible as a sum of two distinct primes in exactly n ways.

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