A333779 Positive numbers m used to build entire prime set by m +/- n without duplication or 0 if there is no such m.

2, 4, 9, 16, 27, 42, 23, 60, 51, 70, 93, 120, 85, 114, 153, 56, 165, 174, 155, 132, 213, 218, 201, 234, 253, 288, 225, 254, 135, 360, 323, 342, 315, 274, 303, 384, 395, 420, 405, 440, 357, 420, 481, 534, 465, 454

Offset: 0

Author

Marcin Barylski, Apr 05 2020

Keywords

• nonn

Comments

Conjecture: every prime is eventually constructed by the sequence.

Taking into account first 10 terms: a(0),a(1),...a(9) = [2, 4, 9, 16, 27, 42, 23, 60, 51, 70] it is possible to build the following primes: [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 43, 47, 53, 59, 61, 67, 79], the only not covered (yet) primes <= 79 are: [41, 71, 73]. 73 will be covered by a(12)=85 (73=85-12), and both 41 and 71 by a(15)=56 (41=56-15, 71=56+15).

The truth of Polignac's conjecture would imply that all terms are well defined. - Rémy Sigrist, Apr 26 2020

a(n) > 0 for 1 <= n <= 10^6. - David A. Corneth, Jun 06 2020

Examples

			a(0)=2, because 2=2+0=2-0 and 2 is prime.
a(1)=4, because 3=4-1, 5=4+1, both 3 and 5 are primes, not covered yet.
a(1) is not 3 because 3+1=4 is not a prime number.
a(2)=9, because 7=9-2, 11=9+2, both 7 and 11 are primes, not covered yet.
a(2) is not 5 (although 5-2=3 and 5+2=7, both are primes) because 3 is already covered by a term a(1) - this sequence is without duplication.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..10000
• Marcin Barylski, C++ program for generating A333779
• Marcin Barylski, On the symmetry of primes

Crossrefs

Cf. A000040, A000230, A020483.

Programs

Mathematica

Nest[Function[{t, i}, Append[t, Block[{k = 2, s}, While[! AllTrue[Set[s, k + i {-1, 1}], And[PrimeQ@ #, FreeQ[t[[All, -1]], #] ] &], k++]; {k, s}] ]] @@ {#, Length@ #} &, {{2, {2}}}, 60][[All, 1]] (* Michael De Vlieger, May 03 2020 *)

PARI

{ p=2; pp=[]; for (n=0,  45, for (k=1, oo, while (#pppp[#pp], pp = concat(pp, p); p = nextprime(p+1);); if (setsearch(pp, pp[k]+2*n), print1 (pp[k]+n", "); pp = setminus(pp, Set([pp[k], pp[k]+2*n])); break))) } \\ Rémy Sigrist, Jun 06 2020

Extensions

More terms from Michael De Vlieger, May 03 2020