A054860 -id:A054860 - cp's OEIS frontend

A068307 From Goldbach problem: number of decompositions of n into a sum of three primes.

0, 0, 0, 0, 0, 1, 1, 1, 2, 1, 2, 2, 2, 1, 3, 2, 4, 2, 3, 2, 5, 2, 5, 3, 5, 3, 7, 3, 7, 2, 6, 3, 9, 2, 8, 4, 9, 4, 10, 2, 11, 3, 10, 4, 12, 3, 13, 4, 12, 5, 15, 4, 16, 3, 14, 5, 17, 3, 16, 4, 16, 6, 19, 3, 21, 5, 20, 6, 20, 2, 22, 5, 21, 6, 22, 5, 28, 5, 24, 7

Offset: 1

Naohiro Nomoto, Feb 24 2002

For even n > 2, a(n) = A061358(n-2). - Reinhard Zumkeller, Aug 08 2009
Vinogradov proved that every sufficiently large odd number is the sum of three primes. - T. D. Noe, Mar 27 2013
The two Helfgott papers show that every odd number greater than 5 is the sum of three primes (this is the Odd Goldbach Conjecture). - T. D. Noe, May 14 2013, N. J. A. Sloane, May 18 2013

			a(6) = 1 because 6 = 2+2+2,
a(9) = 2 because 9 = 2+2+5 = 3+3+3,
a(15) = 3 because 15 = 2+2+11 = 3+5+7 = 5+5+5,
a(17) = 4 because 17 = 2+2+13 = 3+3+11 = 3+7+7 = 5+5+7.
- _Zak Seidov_, Jun 29 2017

Robert G. Wilson v, Table of n, a(n) for n = 1..36000 (first 10000 terms from T. D. Noe)
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.
Yannick Saouter, Checking the odd Goldbach conjecture up to 10^20, Math. Comp. 67 (222) (1998) 863-866.
Eric Weisstein's World of Mathematics, Vinogradov's Theorem
Wikipedia, Goldbach's conjecture.

Bisections: A045917, A054860. Cf. A002375, A007963, A061358, A059998.
First occurrence: A139321. Records: A139322.
Column k=3 of A117278.

f[n_] := Block[{c = 0, lmt = PrimePi@ Floor[n/2], p, q}, Do[p = Prime@ i; q = Prime@ j; r = n - p - q; If[ PrimeQ@ r && r >= p, c++ ], {i, lmt}, {j, i}]; c]; Array[f, 91] (* Robert G. Wilson v, Apr 13 2008 *)
Table[Count[IntegerPartitions[n,{3}],?(AllTrue[#,PrimeQ]&)],{n,50}] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale, Sep 10 2019 *)

a(n)=my(s); forprime(p=(n+2)\3,n-4, forprime(q=(n-p+1)\2,min(n-p-2,p), if(isprime(n-p-q), s++))); s \\ Charles R Greathouse IV, Jun 29 2017

from sympy import isprime, primerange, floor
def a(n):
    s=0
    for p in primerange(((n + 2)//3), n - 3):
        for q in primerange(((n - p + 1)//2), min(n - p - 2, p) + 1):
            if isprime(n - p - q): s+=1
    return s
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jul 01 2017, after PARI code by Charles R Greathouse IV

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} A010051(i) * A010051(k) * A010051(n-i-k). - Wesley Ivan Hurt, Mar 26 2019

a(n) = [x^n y^3] Product_{k>=1} 1/(1 - y*x^prime(k)). - Ilya Gutkovskiy, Apr 18 2019

More terms from Vladeta Jovovic, Mar 10 2002

A083338 Number of partitions of odd numbers into three primes and of even numbers into two primes.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 1, 2, 2, 2, 1, 2, 2, 3, 2, 4, 2, 3, 2, 5, 3, 5, 3, 5, 3, 7, 2, 7, 3, 6, 2, 9, 4, 8, 4, 9, 2, 10, 3, 11, 4, 10, 3, 12, 4, 13, 5, 12, 4, 15, 3, 16, 5, 14, 3, 17, 4, 16, 6, 16, 3, 19, 5, 21, 6, 20, 2, 20, 5, 22, 6, 21, 5, 22, 5, 28, 7, 24, 4, 25, 5, 29, 8, 27, 5, 29, 4, 33, 9, 29, 4

Offset: 1

Reinhard Zumkeller, Apr 24 2003

a(n) > 0 for all n iff Goldbach's conjectures hold.

Antti Karttunen, Table of n, a(n) for n = 1..1001
Antti Karttunen, Scheme-program for computing A083338 and A083339
Eric Weisstein's World of Mathematics, Goldbach Conjecture.
Index entries for sequences related to Goldbach conjecture

Cf. A000607, A061358, A068307, A071335, A083339.

f[n_] := Length@ IntegerPartitions[n, If[ OddQ@ n, {3}, {2}], Prime@ Range@ PrimePi@ n]; Array[f, 92] (* Robert G. Wilson v, Nov 28 2012 *)

a(n) = if n is even then A045917(n/2) else A054860((n-1)/2).

For even n: a(n) = A061358(n); for odd n: a(n) = A068307(n). - Antti Karttunen, Sep 14 2017

A102605 Number of ways of writing 2n+1 as p+q+r where p,q,r are primes with p < q < r, offset=0.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 6, 6, 9, 8, 8, 11, 11, 10, 13, 13, 12, 14, 15, 13, 18, 17, 14, 21, 19, 17, 25, 20, 21, 26, 25, 22, 30, 28, 21, 32, 31, 23, 37, 32, 27, 39, 36, 32, 43, 41, 36, 45, 44, 35, 51, 48, 34, 54, 48, 36, 59, 50, 43, 60, 55, 46, 61

Offset: 0

Zak Seidov, Jan 29 2005

The graph of this function shows two main branches, each with further subdivisions. It seems that the main branches result from the fact that values a(3k+1) are in the mean roughly 30% lower than values a(3k) and a(3k+2). This can be explained by the fact that the sum of 3 primes (with equal probability of being congruent to 1 or to 5 mod 6) is congruent to 3 (mod 6) in only 2 out of 8 cases, and congruent to 1 or to 5 (mod 6) in 3 out of 8 cases, for each of these two residues. Analyzing the frequencies of the possible residues mod 30 explains the further sub-branches: A sum of 3 primes is congruent to 1, 3, ..., 29 (mod 30) in (42, 29, 33, 39, 29, 36, 36, 30, 39, 30, 39, 30, 36, 37, 27) out of 512 cases. - M. F. Hasler, Oct 27 2017

			a(19) = 6 because 2*19+1 = 39 and 39 = 3+5+31 = 3+7+29 = 3+13+23 = 3+17+19 = 5+11+23 = 7+13+19.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Number of ways of writing 2n+1 as p+q+r where p, q, r are primes with p <= q <= r gives A054860.

Bisection of A125688 (odd part). - Alois P. Heinz, Nov 14 2012

A102605(n,s=0)={forprime(p=1,(n*=2)\3,my(d=n-p);forprime(q=p+1,d\2,isprime(d+1-q)&&s++));s} \\ M. F. Hasler, Oct 27 2017

A288574 Total number of distinct primes in all representations of 2*n+1 as a sum of 3 odd primes.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 4, 4, 6, 7, 9, 10, 12, 15, 17, 16, 19, 19, 23, 25, 26, 26, 28, 33, 32, 35, 43, 39, 41, 45, 45, 48, 54, 55, 52, 60, 59, 56, 75, 67, 67, 81, 74, 76, 92, 83, 85, 100, 96, 81, 106, 103, 91, 121, 108, 98, 131, 120, 116, 143, 133, 129, 151, 144, 124, 163

Offset: 0

Franklin T. Adams-Watters and R. J. Mathar, Jun 28 2017

nonn

That is, a representation 2n+1 = p+p+p counts as 1, as p+p+q counts as 2, and p+q+r counts as 3. If each representation is counted once, we simply get A007963.

Indranil Ghosh (first 200 terms), Hugo Pfoertner, Table of n, a(n) for n = 0..10000

A288573 appears to be an erroneous version of this sequence.

Cf. A007963, A054860, A087916.

A288574 := proc(n)
    local a, i, j, k, p, q, r,pqr ;
    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
                    pqr := {p,q,r} ;
                    a := a+nops(pqr) ;
                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(A288574(n),n=0..80) ; # R. J. Mathar, Jun 29 2017

Table[x = 2 n + 1; max = PrimePi[x]; Total[Length /@ Tally /@ DeleteDuplicates[Sort /@ Select[Tuples[Range[2, max], 3], Prime[#[[1]]] + Prime[#[[2]]] + Prime[#[[3]]] == x &]]], {n, 0, 100}] (* Robert Price, Apr 22 2025 *)

a(n)={my(p,q,r,cnt);n=2*n+1;
forprime(p=3,n\3,forprime(q=p,(n-p)\2,
if(isprime(r=n-p-q), cnt+=if(p===q&&p==r,1,if(p==q||q==r,2,3)))));cnt}
\\ Franklin T. Adams-Watters, Jun 28 2017

from sympy import primerange, isprime
def a(n):
    n=2*n + 1
    c=0
    for p in primerange(3, n//3 + 1):
        for q in primerange(p, (n - p)//2 + 1):
            r=n - p - q
            if isprime(r): c+=1 if p==q and p==r else 2 if p==q or q==r else 3
    return c
print([a(n) for n in range(66)]) # Indranil Ghosh, Jun 29 2017

A083339 a(n) is the number of distinct prime factors of n that occur in partitions into two primes when n is even and into three primes when n is odd.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Apr 24 2003

nonn

Number of distinct prime factors of n that occur in prime-partitions confirming Goldbach's conjectures. (The original name of this sequence.)

Conjecture: Apart from k=2, A070826(k): 1, 3, 15, 105, 1155, 15015, 255255, gives the positions of records (each equal to k-1). This follows from the conjectured formula. - Antti Karttunen, Sep 14 2017

			For n = 14 = 2*7 = 3 + 11 = 7 + 7, only one factor of 14 occurs, thus a(14) = 1.
For n = 15 = 3*5 = 2 + 2 + 11 = 3 + 5 + 7 = 5 + 5+ 5, both factors of 15 occur, thus a(15) = 2.
For n = 105 = 3*5*7, with 35 different partitions into three primes, the partition 97 + 5 + 3 contains the prime factors 3 and 5, while the partition 79 + 19 + 7 contains 7, thus all three prime factors of 115 occur and a(115) = 3.
For n = 1155 = 3*5*7*11, among 891 different partitions into three primes, the following four partitions: 1129 + 23 + 3 = 1129 + 19 + 7 = 1109 + 41 + 5 = 1103 + 41 + 11 each have either 3, 5, 7 or 11 as one of their parts, thus a(1155) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..2048
Antti Karttunen, Scheme-program for computing A083338 and A083339
Eric Weisstein's World of Mathematics, Goldbach Conjecture.
Index entries for sequences related to Goldbach conjecture

Cf. A001221, A083338, A045917, A054860, A070826, A100484.

Table[Count[Union@ Flatten@ Select[IntegerPartitions[n, {2 + Boole[OddQ@ n]}], AllTrue[#, PrimeQ] &], p_ /; Divisible[n, p]], {n, 105}] (* Michael De Vlieger, Sep 16 2017 *)

If n is even, a(n) = A010051(n/2), if n is an odd prime, a(n) = 0, and for odd composites (conjecturally), a(n) = A001221(n). - Antti Karttunen, Sep 14 2017

Name edited and two further examples added by Antti Karttunen, Sep 14 2017

A218797 Number of ways to write 2n - 1 as p + q + r with p <= q <= r and p, q, r, p^2 + q^2 + r^2 all prime.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Nov 05 2012

nonn

Conjecture: a(n) > 0 for all n=1715,1716,....

This conjecture is stronger than the weak Goldbach conjecture. It has been verified for n up to 500,000. Those 0

			a(7)=2 since 13=3+3+7=3+5+5, and both 3^2+3^2+7^2=67 and 3^2+5^2+5^2=59 are primes.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000040, A054860, A085317, A218754, A218585, A218654, A218656.

a[n_]:=a[n]=Sum[If[PrimeQ[n-Prime[j]-Prime[k]]==True&&PrimeQ[Prime[j]^2+Prime[k]^2+(n-Prime[j]-Prime[k])^2]==True,1,0],{j,1,PrimePi[n/3]},{k,j,PrimePi[(n-Prime[j])/2]}]
Do[Print[n," ",a[2n-1]],{n,1,10000}]

Showing 1-6 of 6 results.

cp's OEIS Frontend

A068307 From Goldbach problem: number of decompositions of n into a sum of three primes.

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A083338 Number of partitions of odd numbers into three primes and of even numbers into two primes.

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A102605 Number of ways of writing 2n+1 as p+q+r where p,q,r are primes with p < q < r, offset=0.

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A288574 Total number of distinct primes in all representations of 2*n+1 as a sum of 3 odd primes.

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A083339 a(n) is the number of distinct prime factors of n that occur in partitions into two primes when n is even and into three primes when n is odd.

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A218797 Number of ways to write 2n - 1 as p + q + r with p <= q <= r and p, q, r, p^2 + q^2 + r^2 all prime.

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