A278923 -id:A278923 - cp's OEIS frontend

A278922 Largest p such that n = p + q + r where p < q < r are all prime, or 0 if no such primes p, q, r exist.

0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 2, 3, 2, 0, 2, 3, 2, 3, 2, 5, 2, 5, 2, 3, 2, 5, 2, 7, 2, 5, 2, 7, 2, 7, 2, 7, 2, 11, 2, 11, 2, 5, 2, 11, 2, 13, 2, 11, 2, 13, 2, 13, 2, 11, 2, 17, 2, 13, 2, 13, 2, 17, 2, 17, 2, 17, 2, 19, 2, 19, 2, 13, 2, 17, 2, 19, 2, 17, 2, 23, 2, 19, 2, 19, 2, 23, 2, 23, 2, 23, 2, 23, 2, 29, 2, 23, 2, 29, 2, 29, 2, 23, 2, 29, 2, 31, 2, 31, 2, 29, 2, 31, 2, 29, 2, 31, 2, 37

Offset: 1

M. F. Hasler, Dec 01 2016

nonn

Empirically, a(n) >= 2 for all n >= 18. Since a(2n) = 2 unless it is zero, the terms with even indices are less interesting, and the terms with odd indices are listed in A278923.
For even n, the existence of the three primes reduces to a slightly strengthened* variant of Goldbach's conjecture. For odd n, is a slightly strengthened* variant of the weak (a.k.a. odd, or ternary) Goldbach conjecture, considered to be proved since 2013. (*) In both cases, the strengthening consists of requiring that the three primes must be distinct.
From Robert G. Wilson v, Dec 02 2016: (Start)
The first occurrence of the n-th prime: 10, 15, 23, 31, 41, 49, 59, 71, 83, 97, 109, 121, 131, 143, 159, 173, 187, 199, 211, 223, 235, 251, 269, 287, 301, 311, 319, 329, 349, 371, 395, 407, 425, 439, 457, 471, 487, 503, ..., .
Conjecture: primes appear in their natural order. (End)

David W. Wilson, Table of n, a(n) for n = 1..10000
Index entries for sequences related to Goldbach conjecture

Cf. A278923.

Cf. A278373, complement of A056996.

f[n_] := If[OddQ@n || n < 18, Block[{p = 0, q = 3, r = 5}, While[q < r, r = NextPrime@ q; While[r < n - q - 1, If[n < 2q + r && PrimeQ[n - r - q], p = Max[p, n - r - q]; Break[]]; r = NextPrime@ r]; q = NextPrime@ q]; p], 2]; Array[f, 121] (* Robert G. Wilson v, Dec 02 2016 *)

a(n,p=if(bittest(n,0),n\3-1,3))=while(p=precprime(p-1),forprime(q=p+1,(n-p-1)\2,isprime(n-p-q)&&return(p)))

a(2n) = 2 (for n > 4), since one of the three primes must necessarily be even, and that can only be p = 2.

a(n) = 0 for n < 2 + 3 + 5 = 10, and for odd n < 3 + 5 + 7 = 15.

cp's OEIS Frontend

A278922 Largest p such that n = p + q + r where p < q < r are all prime, or 0 if no such primes p, q, r exist.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula