A139321 -id:A139321 - 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

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.

A139322 Record values of n in A068307.

Original entry on oeis.org

1, 6, 9, 15, 17, 21, 27, 33, 39, 41, 45, 47, 51, 53, 57, 63, 65, 71, 77, 83, 89, 95, 101, 107, 113, 119, 125, 131, 137, 143, 149, 155, 161, 167, 173, 185, 191, 197, 203, 209, 215, 221, 227, 233, 239, 245, 251, 257, 269, 275, 277, 281, 287, 293, 299, 305, 311, 317

Offset: 0

Robert G. Wilson v, Apr 13 2008

nonn

Robert G. Wilson v and T. D. Noe, Table of n, a(n) for n = 0..5000

Cf. A000040, A068307, A139321.

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]; lst = {1}; Do[ If[f@n > f@lst[[ -1]], AppendTo[lst, n]], {n, 13450}]; lst

A347436 a(n) is the least odd number that has exactly n decompositions as the sum of three primes, or 0 if there is no such odd number.

Original entry on oeis.org

1, 7, 9, 15, 17, 21, 31, 27, 35, 33, 39, 41, 45, 47, 55, 51, 53, 57, 0, 63, 67, 65, 71, 0, 79, 81, 0, 85, 77, 83, 99, 0, 0, 89, 97, 95, 103, 111, 101, 0, 0, 0, 115, 107, 0, 129, 121, 113, 0, 141, 119, 0, 0, 125, 133, 147, 0, 131, 159, 145, 153, 151, 137, 0, 0, 143, 0, 0, 149, 155, 0, 0, 0, 163, 189

Offset: 0

J. M. Bergot and Robert Israel, Sep 02 2021

nonn

Entries of 0 are conjectural. If nonzero they are greater than 10^5.

Assuming Goldbach's conjecture, if k is odd then A068307(k) >= pi(k-4)-pi((k-1)/2). Using Pierre Dusart's bounds on pi(x), this implies that, for example, A068307(k) >= 4292 for odd k >= 10^5. Thus (on the assumption of Goldbach's conjecture) the given entries of 0 are correct.

			a(3) = 15 because 15 has exactly 3 decompositions as the sum of 3 primes: 2+2+11 = 3+5+7 = 5+5+5, and it is the smallest odd number that does.

Cf. A068307, A139321.

N:= 10^5:
P:= select(isprime, [2,seq(i,i=3..N,2)]):
nP:=nops(P):
V:= Vector(N):
for i from 1 to nP do
  for j from i to nP while P[i]+P[j] <= N do
    for k from j to nP do
      n:= P[i]+P[j]+P[k];
      if n > N then break fi;
      V[n]:= V[n]+1;
od od od:
R:= Vector(300):
for i from 1 to N by 2 do
  if V[i] <= 300 and V[i] > 0 and R[V[i]] = 0 then R[V[i]]:= i fi
od:
convert(R,list);

cp's OEIS Frontend

A068307 From Goldbach problem: number of decompositions of n into a sum of three 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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A139322 Record values of n in A068307.

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A347436 a(n) is the least odd number that has exactly n decompositions as the sum of three primes, or 0 if there is no such odd number.

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