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

Author

Walter Kehowski, Jan 10 2007

Keywords

• nonn

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

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

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.

Examples

			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,...
		

Links

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

Crossrefs

Cf. A119751, A119752, A127256.

Programs

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

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