A127257 a(n) is the initial element of the sequence A(n) defined exactly like A119752 but with the additional condition that each of its elements must not be contained in any of the sequences A(k) for k < n.

2, 6, 18, 20, 50, 68, 74, 78, 90, 96, 134, 138, 156, 200, 204, 228, 278, 288, 296, 306, 326, 338, 378, 384, 398, 404, 440, 464, 468, 504, 510, 524, 530, 546, 600, 608, 644, 660, 704, 726, 740, 774, 828, 854, 870, 930, 938, 944, 966, 986, 1008, 1034, 1068

Offset

1,1

Comments

Recall that A119752 is the sequence defined recursively by a(1)=2 and a(k) is the first even number greater than a(k-1) such that 2a(k)+1 is prime and a(k)+a(j)+1 is prime for all 1<=j<k.

a(n)=A(n,1), the first element of each sequence A(n) defined recursively as follows. Let A(1)=A119752, that is, A(1,k)=A119752(k). Then A(n) is the sequence defined recursively as follows: (1) A(n,1) is the first even number not in any A(m), 1<=m<n, such that 2A(n,1)+1 is prime. (2) A(n,k) is the first even number greater than A(n,k-1), not in any A(m), 1<=m<n, such that 2A(n,k)+1 is prime. (3) A(n,k)+A(n,j)+1 is prime for all 1<=j<k.

Let us say that a positive integer t is eventually in A(n) if 2^p*t, p>=1, is in A(n). For example, if t=19, then A(36,1)=2^5*19. The only numbers not eventually in some A(n) are all powers 2^p such that 2^(p+1)+1 is not prime, (so p is not 0, 1, 3, 7, 15) and Sierpinski numbers of the second kind, namely odd integers t such that 2^p*t+1 is composite for all p>=1. The smallest known example is t=78,557.

Links

Table of n, a(n) for n=1..53.

Eric Weisstein's World of Mathematics, Sierpinski Number of the Second Kind.

Example

a(1)=2 is the first element of A119752=2, 8, 14, 44, 224, 638, 1274, 4004, 675404, ... so a(2)=6 since 6 is the first even number not in A119752 such that 2*6+1 is prime. Furthermore, A(2)=6, 30, 36, 120, 300, 426, 8700, 71910,...

Prog

(PARI) isok(va, k, n) = if (isprime(2*k+1), for (i=1, n-1, if (! isprime(va[i]+k+1), return(0))); return(1));
avec(start, lim) = my(list=List(), n=2, ok=1); listput(list, start); while(ok, my(k=list[n-1]+2); while (!isok(list, k, n), k+=2; if (k>lim, ok=0; break)); if (ok, listput(list, k)); n++; ); Vec(list);
findfirst(v, lim) = forstep (n=2, lim, 2, if (isprime(2*n+1) && !vecsearch(v, n), return(n)));
lista(lim) = my(list = List()); listput(list, 2); my(v=avec(2, lim)); while (1, my(new = findfirst(v, lim)); if (! new, break); my(w = avec(new, lim)); v = vecsort(concat(v, w)); listput(list, new); ); Vec(list); \\ Michel Marcus, Mar 06 2023

Crossrefs

Cf. A119751, A119752, A127256.

Sequence in context: A097635 A151870 A045962 * A050832 A300600 A342035

Adjacent sequences: A127254 A127255 A127256 * A127258 A127259 A127260

Keyword

nonn

Author

Walter Kehowski, Jan 10 2007

Status

approved