|
| |
|
|
A333779
|
|
Positive numbers m used to build entire prime set by m +/- n without duplication or 0 if there is no such m.
|
|
1
|
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,1
|
|
|
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
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
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.
|
|
|
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 *)
|
|
|
PROG
|
(PARI) { p=2; pp=[]; for (n=0, 45, for (k=1, oo, while (#pp<k || pp[k]+2*n>pp[#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
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
