OFFSET
1,1
COMMENTS
Balog shows that a(n) exists for each n. On Dickson's conjecture, Tao & Ziegler show that there is an infinite set S with this property. E. Bombieri showed that a(12) <= 434717 with the set {5, 17, 521, 42281, 138461, 195137, 204137, 221537, 363497, 367001, 414737, 434717}.
REFERENCES
Antal Balog, Linear equations in primes, Mathematika 39 (1992), no. 2, pp. 367-378.
LINKS
Terence Tao, Infinite partial sumsets in the primes, blog post (2023).
Terence Tao and Tamar Ziegler, Infinite partial sumsets in the primes, arXiv preprint, arXiv:2301.10303 [math.NT], 2023.
EXAMPLE
Examples of such sets:
1: { 2}
2: { 3, 7}
3: { 5, 7, 11}
4: { 5, 13, 17, 23}
5: { 5, 11, 31, 41, 47}
6: {11, 29, 41, 59, 67, 71}
7: {17, 23, 29, 43, 83, 149, 233}
8: {23, 29, 43, 83, 149, 167, 233, 419}
9: {11, 29, 59, 71, 167, 211, 389, 431, 449}
10: {17, 23, 83, 113, 149, 349, 569, 659, 863, 1409}
11: {17, 23, 83, 113, 149, 349, 569, 659, 863, 1409, 1889}
12: {29, 239, 1019, 1061, 1571, 1637, 2309, 2699, 3529, 4019, 4211, 4649}
13: {19, 53, 263, 719, 1229, 1433, 1973, 4073, 5849, 6257, 6863, 9623, 11933}
14: {239, 1733, 1847, 2069, 4133, 5279, 10193, 10753, 11393, 20399, 21599, 25169, 25703, 39953}
PROG
(PARI) find(n, mx=oo, v=[])=if(n==0, return(v)); forprime(p=2, mx, for(i=1, #v, if(!isprime(v[i]+p+1), next(2))); my(t=find(n-1, precprime(p-1), concat(p, v))); if(t, return(t))); 0
a(n)=if(n>1, find(n)[n], 2)
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Charles R Greathouse IV, Apr 28 2023
EXTENSIONS
Corrected by Jean-Marc Rebert, May 02 2023
a(13)-a(14) from Martin Ehrenstein, May 04 2023
STATUS
approved