A068307 -id:A068307 - cp's OEIS frontend

A051034 Minimal number of primes needed to sum to n.

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

Offset: 2

Eric W. Weisstein

			a(2) = 1 because 2 is already prime.
a(4) = 2 because 4 = 2+2 is a partition of 4 into 2 prime parts and there is no such partition with fewer terms.
a(27) = 3 because 27 = 3+5+19 is a partition of 27 into 3 prime parts and there is no such partition with fewer terms.

T. D. Noe, Table of n, a(n) for n = 2..10000
Olivier Ramaré, On Šnirel'man's constant, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4e série, 22:4 (1995), pp. 645-706.
Yannick Saouter, Vinogradov's theorem is true up to 10^20
Terence Tao, Every odd number greater than 1 is the sum of at most five primes, arXiv:1201.6656 [math.NT], 2012 preprint, to appear in Mathematics of Computation.
Eric Weisstein's World of Mathematics, Prime Partition
Index entries for sequences related to Goldbach conjecture

Cf. A025583, A004526, A007944, A007962, A000607, A051036, A010051, A061358, A068307, A103765.

Different from A072491.

(* Assuming Goldbach's conjecture *) a[p_?PrimeQ] = 1; a[n_] := If[ Reduce[ n == x + y, {x, y}, Primes] === False, 3, 2]; Table[a[n], {n, 2, 112}] (* Jean-François Alcover, Apr 03 2012 *)

issum(n,k)=if(k==1,isprime(n),k--;forprime(p=2,n,if(issum(n-p,k),return(1)));0)
a(n)=my(k);while(!issum(n,k++),);k \\ Charles R Greathouse IV, Jun 01 2011

a(n) = 1 iff n is prime. a(2n) = 2 (for n > 1) if Goldbach's conjecture is true. a(2n+1) = 2 (for n >= 1) if 2n+1 is not prime, but 2n-1 is. a(2n+1) >= 3 (for n >= 1) if both 2n+1 and 2n-1 are not primes (for sufficiently large n, a(2n+1) = 3 by Vinogradov's theorem, 1937). - Franz Vrabec, Nov 30 2004
a(n) <= 3 for all n, assuming the Goldbach conjecture. - N. J. A. Sloane, Jan 20 2007
a(2n+1) <= 5, see Tao 2012. - Charles R Greathouse IV, Jul 09 2012
Assuming Goldbach's conjecture, a(n) <= 3. In particular, a(p)=1; a(2*n)=2 for n>1; a(p+2)=2 provided p+2 is not prime; otherwise a(n)=3. - Sean A. Irvine, Jul 29 2019
a(2n+1) <= 3 by Helfgott's proof of Goldbach's ternary conjecture, and hence a(n) <= 4 in general. - Charles R Greathouse IV, Oct 24 2022

More terms from Naohiro Nomoto, Mar 16 2001

A125688 Number of partitions of n into three distinct primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 1, 2, 2, 2, 2, 2, 2, 3, 3, 2, 3, 2, 4, 3, 4, 2, 5, 3, 5, 4, 6, 1, 6, 3, 6, 4, 6, 3, 9, 3, 8, 5, 8, 4, 11, 3, 11, 5, 10, 3, 13, 3, 13, 6, 12, 2, 14, 5, 15, 6, 13, 2, 18, 5, 17, 6, 14, 4, 21, 5, 19, 7, 17, 4, 25, 4, 20, 8, 21, 4, 26, 4, 25, 9, 22, 4

Offset: 1

Reinhard Zumkeller, Nov 30 2006

a(A124868(n)) = 0; a(A124867(n)) > 0;

a(A125689(n)) = n and a(m) <> n for m < A125689(n).

			a(42) = #{2+3+37, 2+11+29, 2+17+23} = 3.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Reinhard Zumkeller)

Column k=3 of A219180. - Alois P. Heinz, Nov 13 2012

Cf. A010051, A068307, A124868, A125689.

b:= proc(n, i) option remember; `if`(n=0, [1,0$3], `if`(i<1, [0$4],
       zip((x, y)->x+y, b(n, i-1), [0, `if`(ithprime(i)>n, [0$3],
       b(n-ithprime(i), i-1)[1..3])[]], 0)))
    end:
a:= n-> b(n, numtheory[pi](n))[4]:
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 15 2012

b[n_, i_] := b[n, i] = If[n == 0, {1, 0, 0, 0}, If[i<1, {0, 0, 0, 0}, Plus @@ PadRight[{b[n, i-1], Join[{0}, If[Prime[i]>n, {0, 0, 0}, Take[b[n-Prime[i], i-1], 3]]]}]]]; a[n_] := b[n, PrimePi[n]][[4]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 30 2014, after Alois P. Heinz *)
dp3Q[{a_,b_,c_}]:=Length[Union[{a,b,c}]]==3&&AllTrue[{a,b,c},PrimeQ]; Table[ Count[IntegerPartitions[n,{3}],?dp3Q],{n,100}] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale, Jan 30 2019 *)

a(n)=my(s);forprime(p=n\3,n-4,forprime(q=(n-p)\2+1,min(n-p,p-1),if(isprime(n-p-q),s++)));s \\ Charles R Greathouse IV, Aug 27 2012

From Alois P. Heinz, Nov 22 2012: (Start)

G.f.: Sum_{0

a(n) = [x^n*y^3] Product_{i>=1} (1+x^prime(i)*y). (End)

a(n) = Sum_{k=1..floor((n-1)/3)} Sum_{i=k+1..floor((n-k-1)/2)} A010051(i) * A010051(k) * A010051(n-i-k). - Wesley Ivan Hurt, Mar 29 2019

A230219 Number of ways to write 2*n + 1 = p + q + r with p <= q such that p, q, r are primes in A230217 and p + q + 9 is also prime.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 1, 2, 5, 2, 2, 4, 3, 1, 4, 3, 1, 4, 1, 2, 5, 2, 3, 4, 3, 3, 8, 6, 3, 12, 6, 2, 13, 3, 3, 7, 6, 4, 5, 4, 4, 8, 7, 4, 12, 7, 3, 19, 6, 3, 16, 5, 4, 9, 5, 5, 7, 10, 4, 5, 8, 3, 14, 4, 3, 14, 2, 5, 12, 5, 2, 14, 9, 2, 10, 12, 4, 12, 7, 6, 12, 7, 9, 14, 8, 6, 12, 5, 4, 19, 8, 4, 23, 6, 3, 14

Offset: 1

Zhi-Wei Sun, Oct 11 2013

nonn

Conjecture: a(n) > 0 for all n > 6.
This implies Goldbach's weak conjecture for odd numbers and also Goldbach's conjecture for even numbers.
The conjecture also implies that there are infinitely many primes in A230217.

			a(18) = 1 since 2*18 + 1 = 7 + 13 + 17, and 7, 13, 17 are terms of A230217, and 7 + 13 + 9 = 29 is a prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.

Cf. A002375, A068307, A230217, A230140, A230141.

RQ[n_]:=PrimeQ[n+6]&&PrimeQ[3n+8]
SQ[n_]:=PrimeQ[n]&&RQ[n]
a[n_]:=Sum[If[RQ[Prime[i]]&&RQ[Prime[j]]&&PrimeQ[Prime[i]+Prime[j]+9]&&SQ[2n+1-Prime[i]-Prime[j]],1,0],{i,1,PrimePi[n-1]},{j,i,PrimePi[2n-2-Prime[i]]}]
Table[a[n],{n,1,100}]

A007963 Number of (unordered) ways of writing 2n+1 as a sum of 3 odd primes.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 6, 8, 7, 9, 10, 10, 10, 11, 12, 12, 14, 16, 14, 16, 16, 16, 18, 20, 20, 20, 21, 21, 21, 27, 24, 25, 28, 27, 28, 33, 29, 32, 35, 34, 30, 37, 36, 34, 42, 38, 36, 46, 42, 42, 50, 46, 47, 53, 50, 45, 56, 54, 46, 62, 53, 48, 64, 59, 55, 68, 61, 59, 68

Offset: 0

R. Muller

nonn

Ways of writing 2n+1 as p+q+r where p,q,r are odd primes with p <= q <= r.

The two papers of Helfgott appear to provide a proof of the Odd Goldbach Conjecture that every odd number greater than five is the sum of three primes. (The paper is still being reviewed.) - Peter Luschny, May 18 2013; N. J. A. Sloane, May 19 2013

			a(10) = 4 because 21 = 3+5+13 = 3+7+11 = 5+5+11 = 7+7+7.

George E. Andrews, Number Theory (NY, Dover, 1994), page 111.
Ivars Peterson, The Mathematical Tourist (NY, W. H. Freeman, 1998), pages 35-37.
Paulo Ribenboim, "VI, Goldbach's famous conjecture," The New Book of Prime Number Records, 3rd ed. (NY, Springer, 1996), pages 291-299.

T. D. Noe, Table of n, a(n) for n = 0..10000
H. A. Helfgott, Minor arcs for Goldbach's problem, arXiv:1205.5252 [math.NT], 2012.
H. A. Helfgott, Major arcs for Goldbach's theorem, arXiv:1305.2897 [math.NT], 2013.
H. A. Helfgott, The ternary Goldbach conjecture is true, arxiv:1312.7748 [math.NT], 2013.
H. A. Helfgott, The ternary Goldbach problem, arXiv:1404.2224 [math.NT], 2014.
F. Smarandache, Only Problems, Not Solutions!.
Index entries for sequences related to Goldbach conjecture

Cf. A068307, A087916, A294294 (lower bound of scatterplot), A294357, A294358 (records).

A007963 := proc(n)
    local a,i,j,k,p,q,r ;
    a := 0 ;
    for i from 2 do
        p := ithprime(i) ;
        for j from i do
            q := ithprime(j) ;
            for k from j do
                r := ithprime(k) ;
                if p+q+r = 2*n+1 then
                    a := a+1 ;
                elif p+q+r > 2*n+1 then
                    break;
                end if;
            end do:
            if p+2*q > 2*n+1 then
                break;
            end if;
        end do:
        if 3*p > 2*n+1 then
            break;
        end if;
    end do:
    return a;
end proc:
seq(A007963(n),n=0..30) ; # R. J. Mathar, Sep 06 2014

nn = 75; ps = Prime[Range[2, nn + 1]]; c = Flatten[Table[If[i >= j >= k, i + j + k, 0], {i, ps}, {j, ps}, {k, ps}]]; Join[{0, 0, 0, 0}, Transpose[Take[Rest[Sort[Tally[c]]], nn+2]][[2]]] (* T. D. Noe, Apr 08 2014 *)

a(n)=my(k=2*n+1,s,t); forprime(p=(k+2)\3,k-6, t=k-p; forprime(q=t\2,min(t-3,p), if(isprime(t-q), s++))); s \\ Charles R Greathouse IV, Mar 20 2017

use ntheory ":all"; sub a007963 { my($n,$c)=(shift,0); forpart { $c++ if vecall { is_prime($) } @; } $n,{n=>3,amin=>3}; $c; }
say "$ ",a007963(2*$+1) for 0..100; # Dana Jacobsen, Mar 19 2017

def A007963(n):
    c = 0
    for p in Partitions(n, length = 3):
        b = True
        for t in p:
            b = is_prime(t) and t > 2
            if not b: break
        if b : c = c + 1
    return c
[A007963(2*n+1) for n in (0..77)]   # Peter Luschny, May 18 2013

Corrected and extended by David W. Wilson

A071335 Number of partitions of n into sum of at most three primes.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 3, 2, 3, 3, 3, 3, 4, 3, 4, 4, 5, 4, 5, 4, 6, 5, 6, 6, 6, 6, 7, 5, 8, 5, 8, 5, 10, 6, 8, 8, 10, 6, 11, 5, 12, 7, 12, 7, 13, 7, 14, 9, 13, 9, 15, 7, 17, 8, 15, 8, 17, 7, 17, 10, 18, 9, 20, 8, 21, 11, 21, 8, 21, 7, 23, 11, 23, 11, 23, 10, 28, 12, 25, 11, 26

Offset: 1

Reinhard Zumkeller, May 19 2002

a(n) = A010051(n) + A061358(n) + A068307(n). [From Reinhard Zumkeller, Aug 08 2009]

			a(21)=6 as 21 = 2+19 = 2+2+17 = 3+5+13 = 3+7+11 = 5+5+11 = 7+7+7.

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

Cf. A002375, A068307, A025583.

goldbachcount[p1_] := (parts=IntegerPartitions[p1, 3]; count=0; n=1;
  While[n<=Length[parts], If[Intersection[Flatten[PrimeQ[parts[[n]]]]][[1]]==True, count++]; n++]; count); Table[goldbachcount[i], {i, 1, 100}] (* Frank M Jackson, Mar 25 2013 *)
Table[Length[Select[IntegerPartitions[n,3],AllTrue[#,PrimeQ]&]],{n,90}] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Oct 21 2016 *)

A124867 Numbers that are the sum of 3 distinct primes.

Original entry on oeis.org

10, 12, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81

Offset: 1

Alexander Adamchuk, Nov 11 2006

(Conjecture) Every number n > 17 is the sum of 3 distinct primes. Natural numbers that are not the sum of 3 distinct primes are listed in A124868.

A125688(a(n)) > 0. - Reinhard Zumkeller, Nov 30 2006

			The first three primes are 2, 3, 5, and 2 + 3 + 5 = 10, so 10 is in the sequence. No smaller integer is in the sequence.
5 + 5 + 5 = 15, but note also 3 + 5 + 7 = 15, so 15 is in the sequence.
Although 13 = 3 + 3 + 7 = 3 + 5 + 5, both of those repeat primes, so 13 is not in the sequence.

Eric W. Weisstein, Goldbach conjecture
Wikipedia, Goldbach's conjecture
Wikipedia, Goldbach's weak conjecture

Cf. A124868 (not the sum of 3 distinct primes), A068307, A125688.

threePrimes[n_] := Module[{p, q, r}, {p, q, r} /. Solve[n == p + q + r && p < q < r, {p, q, r}, Primes]];
Reap[For[n = 10, n <= 100, n++, sol = threePrimes[n]; If[MatchQ[sol, {{, , }..}], Print[n, " ", sol[[1]]]; Sow[n]]]][[2, 1]] (* _Jean-François Alcover, Apr 26 2020 *)
has3DistPrimesPart[n_] := Length[Select[IntegerPartitions[n, {3}], Length[Union[#]] == 3 && Union[PrimeQ[#]] == {True} &]] > 0; Select[Range[100], has3DistPrimesPart] (* Alonso del Arte, Apr 26 2020 *)
Union[Total/@Subsets[Prime[Range[20]],{3}]] (* Harvey P. Dale, Feb 06 2024 *)

a(n)=if(n>5,n+12,[10, 12, 14, 15, 16][n]) \\ Charles R Greathouse IV, Aug 26 2011

A283762 Expansion of (x + Sum_{k>=1} x^prime(k))^3.

Original entry on oeis.org

0, 0, 0, 1, 3, 6, 7, 9, 9, 13, 12, 15, 9, 15, 12, 22, 15, 24, 12, 27, 18, 34, 18, 36, 15, 42, 24, 45, 15, 42, 12, 51, 24, 52, 18, 60, 21, 66, 24, 58, 15, 69, 18, 75, 30, 75, 24, 87, 21, 93, 36, 91, 24, 99, 18, 108, 36, 97, 18, 108, 21, 126, 42, 111, 21, 135, 30, 141, 36, 112, 18, 150, 30, 153, 42, 138, 33, 177, 30, 171, 42

Offset: 0

Ilya Gutkovskiy, Mar 16 2017

nonn

Number of ways to write n as an ordered sum of 3 noncomposite numbers (1 together with the primes) (A008578).

a(2k+1) > 0 for all k > 0 (from the ternary Goldbach's conjecture, proved by H. A. Helfgott).

			a(6) = 7 because we have [3, 2, 1], [3, 1, 2], [2, 3, 1], [2, 2, 2], [2, 1, 3], [1, 3, 2] and [1, 2, 3].

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Ilya Gutkovskiy, Extended graphical example

Cf. A008578, A068307, A098238, A124867.

nmax = 80; CoefficientList[Series[(x + Sum[x^Prime[k], {k, 1, nmax}])^3, {x, 0, nmax}], x]

concat([0, 0, 0], Vec((x + sum(k=1, 80, x^prime(k)))^3 + O(x^81))) \\ Indranil Ghosh, Mar 16 2017

G.f.: (x + Sum_{k>=1} x^prime(k))^3.

A341946 Number of partitions of n into 3 primes (counting 1 as a prime).

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 4, 2, 4, 2, 4, 2, 6, 3, 6, 2, 6, 3, 8, 3, 8, 3, 9, 4, 10, 3, 9, 2, 10, 4, 12, 3, 12, 4, 13, 4, 13, 3, 14, 3, 15, 5, 16, 4, 17, 4, 18, 6, 19, 4, 19, 3, 20, 6, 20, 3, 20, 4, 23, 7, 23, 4, 26, 5, 26, 6, 23, 3, 27, 5, 28, 7, 28, 6, 33, 5, 31, 7, 30, 5, 34

Offset: 3

Ilya Gutkovskiy, Feb 24 2021

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

Cf. A008578, A034891, A068307, A341945, A341947, A341948, A341949, A341950, A341951.

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)))(
     `if`(i=0, 1, ithprime(i)))+b(n, i-1))), x, 4)
    end:
a:= n-> coeff(b(n, numtheory[pi](n)), x, 3):
seq(a(n), n=3..83);  # Alois P. Heinz, Feb 24 2021

b[n_, i_] := b[n, i] = Series[If[n == 0, 1,
     If[i < 0, 0, Function[p, If[p > n, 0, x*b[n - p, i]]][
     If[i == 0, 1, Prime[i]]] + b[n, i-1]]], {x, 0, 4}];
a[n_] := Coefficient[b[n, PrimePi[n]], x, 3];
Table[a[n], {n, 3, 83}] (* Jean-François Alcover, Feb 14 2022, after Alois P. Heinz *)

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A259194 Number of partitions of n into four primes.

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A051034 Minimal number of primes needed to sum to n.

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A259195 Number of partitions of n into five primes.

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A125688 Number of partitions of n into three distinct primes.

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A230219 Number of ways to write 2*n + 1 = p + q + r with p <= q such that p, q, r are primes in A230217 and p + q + 9 is also prime.

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A007963 Number of (unordered) ways of writing 2n+1 as a sum of 3 odd primes.

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A071335 Number of partitions of n into sum of at most three primes.

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