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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. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,10

COMMENTS

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.

LINKS

David W. Wilson, Table of n, a(n) for n = 1..10000

Index entries for sequences related to Goldbach conjecture

FORMULA

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.

MATHEMATICA

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 *)

PROG

(PARI) 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)))

CROSSREFS

Cf. A278923.

Cf. A278373, complement of A056996.

Sequence in context: A294614 A063918 A271419 * A163169 A097974 A333753

Adjacent sequences:  A278919 A278920 A278921 * A278923 A278924 A278925

KEYWORD

nonn

AUTHOR

M. F. Hasler, Dec 01 2016

STATUS

approved

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Last modified March 2 19:18 EST 2021. Contains 341756 sequences. (Running on oeis4.)