A007963 -id:A007963 - 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

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

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 2, 3, 4, 3, 5, 5, 5, 7, 7, 6, 9, 8, 9, 10, 11, 10, 12, 13, 12, 15, 16, 14, 17, 16, 16, 19, 21, 20, 20, 22, 21, 22, 28, 24, 25, 29, 27, 29, 33, 29, 33, 35, 34, 30, 38, 36, 35, 43, 38, 37, 47, 42, 43, 50, 46, 47, 53, 50, 45, 57, 54, 47, 62, 53, 49, 65, 59, 55, 68

Offset: 0

James Sellers, May 25 2000

nonn

Every sufficiently large odd number is the sum of three primes (th. by Vinogradov, 1937). Goldbach's conjecture requires three ODD primes and then a(n) > 0 for n > 2 is weaker.

The unconditional theorem was proved by Helfgott (see link below). - T. D. Noe, May 15 2013

			7 = 2 + 2 + 3 so a(3) = 1;
9 = 2 + 2 + 5 = 3 + 3 + 3 so a(4) = 2;
11 = 2 + 2 + 7 = 3 + 3 + 5 so a(5) = 2.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, appendix 3.
Wolfgang Schwarz, Einfuehrung in Methoden und Ergebnisse der Primzahltheorie, Bibliographisches Institut Mannheim, 1969, ch. 7.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000 (up to 1000 from _T. D. Noe_)
H. A. Helfgott, Major arcs for Goldbach's theorem, arXiv 1305.2897 [math.NT], May 14 2013.

Cf. A007963, A288574.

nn = 201; t = Table[0, {(nn + 1)/2}]; pMax = PrimePi[nn]; ps =
Prime[Range[pMax]]; Do[n = ps[[i]] + ps[[j]] + ps[[k]]; If[n <= nn &&
OddQ[n], t[[(n + 1)/2]]++], {i, pMax}, {j, i, pMax}, {k, j, pMax}]; t (* T. D. Noe, May 23 2017 *)
f[n_] := Length@ IntegerPartitions[2n +1, {3}, Prime@ Range@ PrimePi[2n -3]]; Array[f, 75, 0] (* Robert G. Wilson v, Jun 30 2017 *)

first(n)=my(v=vector(n)); forprime(r=3,2*n-3, v[r\2+2]++); forprime(p=3,(2*n+1)\3, forprime(q=p,(2*n+1-p)\2, forprime(r=q,2*n+1-p-q, v[(p+q+r)\2]++))); concat(0, v) \\ Charles R Greathouse IV, May 25 2017

A087916 Number of ordered ways to write 2n+1 as a sum of 3 odd primes.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 6, 7, 9, 12, 16, 18, 21, 27, 30, 30, 34, 36, 42, 46, 48, 48, 51, 63, 60, 64, 81, 75, 76, 87, 87, 90, 102, 105, 97, 117, 114, 105, 144, 129, 126, 159, 141, 145, 177, 162, 160, 195, 186, 153, 207, 201, 171, 237, 210, 187, 255, 234, 222, 279

Offset: 0

Ralf Stephan, Oct 18 2003

T. D. Noe, Table of n, a(n) for n = 0..10000

Cf. A007963 (unordered), A068307 (with 2).

Cf. A087917.

nn = 100; lim = 2*nn + 17; ps = Prime[Range[2, nn + 1]]; t = Table[0, {lim}]; Do[s = i + j + k; If[s <= lim, t[[s]]++], {i, ps}, {j, ps}, {k, ps}]; Take[t, {1, lim, 2}] (* T. D. Noe, Apr 10 2014 *)

for(n=0, 100, t=2*n+1; c=0; for(k=2, t, for(l=2, t, for(m=2, t, tt=prime(k)+prime(l)+prime(m); if(tt>2*n+1, break); if(tt==2*n+1, c=c+1)))); print1(c", "))

Leading zeros added by T. D. Noe, Apr 10 2014

A294294 Conjecturally, all odd numbers greater than a(n) can be represented in more ways by the sum of 3 odd primes p+q+r with p<=q<=r than a(n).

Original entry on oeis.org

7, 11, 15, 19, 23, 25, 31, 35, 37, 43, 45, 49, 55, 61, 63, 69, 75, 79, 81, 85, 87, 91, 99, 105, 111, 117, 129, 135, 141, 147, 159, 165, 171, 177, 195, 201, 207, 219, 225, 231, 237, 255, 261, 267, 279, 285, 291, 297, 309, 315, 321, 339, 345, 351

Offset: 1

Hugo Pfoertner, Oct 27 2017

nonn

The sequence provides numerical evidence of the validity of the ternary Goldbach conjecture, i.e. that every odd number >5 can be written as the sum of 3 primes, now proved by A. Helfgott.
The corresponding minimum numbers of representations are provided in A294295.
Empirically, mod(a(n),6) = 3 for all a(n) > 91 and mod(a(n),30) = 15 for all a(n) > 1281.

			a(1)=7 because all odd numbers > 7 have more representations by sums of 3 odd primes than 7, which has no such representation (A294295(1)=0).
a(2)=11, because all odd numbers > 11 have at least 2 representations p+q+r, e.g. 13=3+3+7=5+5+3 whereas 11=3+3+5 and 9=3+3+3 only have A294295(2)=1 representation.

For references and links see A007963.

Hugo Pfoertner, Table of n, a(n) for n = 1..301
H. A. Helfgott, The ternary Goldbach conjecture is true, arXiv:1312.7748 [math.NT], 2013-2014.

Cf. A007963, A102605, A139321, A294295, A294357, A294358.

A007963(k) > A007963((a(n)-1)/2) for all k > (a(n)-1)/2.

A294357 Smallest odd number that can be expressed in more ways by sums of 3 odd primes p+q+r with p <= q <= r than any smaller odd number.

Original entry on oeis.org

9, 13, 17, 21, 25, 27, 29, 33, 37, 39, 45, 47, 51, 53, 63, 65, 71, 77, 83, 89, 95, 101, 107, 113, 119, 125, 131, 137, 143, 149, 161, 167, 173, 185, 191, 197, 203, 209, 215, 221, 227, 233, 239, 245, 247, 251, 257, 269, 277, 281, 287, 293, 299

Offset: 1

Hugo Pfoertner, Oct 29 2017

nonn

Position of n-th record in A007963 converted to actual odd number for which the record is achieved.
The corresponding records of numbers of representations are provided in A294358.
Empirically mod(a(n),6) = 5 for all a(n) > 63 and mod(a(n),30) != 5 for all a(n) > 425.

Hugo Pfoertner, Table of n, a(n) for n = 1..1113

Cf. A007963, A102605, A139321, A294294, A294295, A294358.

a(1)=9 because 9 = 3+3+3 is the smallest number that can be represented as sum of 3 odd primes.

a(13)=51 because A007963(25) = A007963((51-1)/2) = 14 is the 13th record in A007963.

A294358 Number of ways to write A294357(n) as a sum of 3 odd primes p+q+r, with p>=q>=r.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 18, 20, 21, 27, 28, 33, 35, 37, 42, 46, 50, 53, 56, 62, 64, 68, 76, 77, 89, 91, 95, 101, 106, 110, 115, 122, 124, 129, 132, 133, 134, 142, 154, 157, 160, 170, 175, 186, 187, 190, 197, 209, 210, 212

Offset: 1

Hugo Pfoertner, Oct 29 2017

nonn

Records in A007963.

Hugo Pfoertner, Table of n, a(n) for n = 1..1113

Cf. A007963, A294357.

a(n) = A007963((A294357(n)-1)/2).

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

A294295 Number of ways to write A294294(n) as a sum of 3 odd primes p+q+r, with p>=q>=r.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 14, 16, 18, 20, 21, 24, 25, 27, 28, 29, 30, 34, 36, 42, 45, 46, 48, 55, 58, 60, 64, 68, 72, 81, 85, 88, 90, 93, 101, 107, 110, 119, 122, 128, 134, 142, 143, 145, 150, 161, 162, 169, 179, 181, 195, 196, 215

Offset: 1

Hugo Pfoertner, Oct 28 2017

nonn

Hugo Pfoertner, Table of n, a(n) for n = 1..301

Cf. A007963, A294294.

a(n) = A007963((A294294(n)-1)/2).

A190353 Goldbach conjecture: number of decompositions of n into an unordered sum of two odd primes (if n even) or three odd primes (if n odd).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 1, 1, 2, 2, 2, 2, 3, 2, 3, 2, 4, 3, 4, 3, 5, 3, 6, 2, 7, 3, 6, 2, 8, 4, 7, 4, 9, 2, 10, 3, 10, 4, 10, 3, 11, 4, 12, 5, 12, 4, 14, 3, 16, 5, 14, 3, 16, 4, 16, 6, 16, 3, 18, 5, 20, 6, 20, 2, 20, 5, 21, 6, 21, 5, 21, 5, 27, 7, 24

Offset: 1

Daniel Forgues, May 09 2011

nonn

This sequence differs from A083338 because A083338 allows 2 as a prime.

Daniel Forgues, Table of n, a(n) for n = 1..20000
OEIS Wiki, Goldbach conjecture

Cf. A002375, A007963, A083338.

a(2n) = A002375(n) and a(2n+1) = A007963(n).

A112418 Primes which have a prime number of partitions into five distinct primes.

Original entry on oeis.org

53, 59, 67, 83, 113, 151, 157, 211, 239, 601, 809, 821, 881, 971, 1237, 1297, 1427, 1669, 1759, 1973, 2069, 2129, 2243, 2333, 2659, 2677, 2719, 2789, 2803, 2999, 3329, 3613, 3623, 3769, 3797, 4001, 4451

Offset: 1

Giorgio Balzarotti and Paolo P. Lava, Dec 09 2005

nonn

The corresponding numbers of partitions are 2,5,11,29,109,331,379,1091...

			53 is there because there are 2 partitions of 53 (3+7+11+13+19, 5+7+11+13+17) and 2 is prime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..102

Cf. A000009, A000041, A007963, A051034.

part5_prime:=proc(N) s:=1; for n from 2 to N do cont:=0; for i from 1 to n-5 do for j from i+1 to n-4 do for k from j+1 to n-3 do for l from k+1 to n-2 do for m from l+1 to n-1 do if(ithprime(n)= ithprime(i)+ithprime(j)+ithprime(k)+ithprime(l)+ithprime(m) then cont:=cont+1; fi; od; od; od; od; od; if (isprime(cont)=true) then a[s]:=ithprime(n); s:=s+1; fi; od; end:

has(n)=my(t,Q,R,S);forprime(p=n\5+1,n-26, Q=n-p; forprime(q=Q\4+1,min(p-1,Q-15), R=Q-q; forprime(r=R\3+1,min(q-1,R-8), S=R-r; forprime(s=S-r+1,(S-1)\2, isprime(S-s) && t++)))); isprime(t)
select(has, primes(100)) \\ Charles R Greathouse IV, Apr 22 2015

list(lim)=my(v=vectorsmall(precprime(lim)),u=List(),Q,R,S); forprime(p=13,#v-26, Q=#v-p; forprime(q=11,min(p-1,Q-15), R=Q-q; forprime(r=7,min(q-1,R-8), S=R-r; forprime(s=5,min(S-2,r-1), forprime(t=3,min(S-s,s-1), v[p+q+r+s+t]++))))); forprime(p=2,lim, if(isprime(v[p]), listput(u,p))); Set(u) \\ Charles R Greathouse IV, Apr 22 2015

Edited by Don Reble, Jan 26 2006

a(31)-a(37) from Charles R Greathouse IV, Apr 22 2015

cp's OEIS Frontend

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

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

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A087916 Number of ordered ways to write 2n+1 as a sum of 3 odd primes.

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A294294 Conjecturally, all odd numbers greater than a(n) can be represented in more ways by the sum of 3 odd primes p+q+r with p<=q<=r than a(n).

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A294357 Smallest odd number that can be expressed in more ways by sums of 3 odd primes p+q+r with p <= q <= r than any smaller odd number.

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A294358 Number of ways to write A294357(n) as a sum of 3 odd primes p+q+r, with p>=q>=r.

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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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A294295 Number of ways to write A294294(n) as a sum of 3 odd primes p+q+r, with p>=q>=r.

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