A307873 The concatenation pkp is the number obtained by placing prime p either side of R_k, the k-th repunit (1, k times); a(n) is the smallest k such that pkp is prime, where p=prime(n), or -1 if no such k exists.

-1, 1, -1, 10905, 15, 2, 1, 2, 3, 1, 3, 173, 1, 14, 1, 43, 1, 5, 11, 1, 2, 3, 3, 1, 2, -1, 5, 421, 3, 1, -1, 1, 1, 3, -1, 15, -1, 3, 3, 163, -1, 3, 13, -1, 679, -1, 5, 5, -1, 107, 93, 1, -1, 3, -1, 1, -1, 9, 5, -1, -1, 9, 1089, -1, 3, 7, 3, 15, -1, 27, -1, 1, -1, 27, 17, 25, 1, 15, 3

Offset

1,4

Comments

Primes (from above data) for which pkp is composite for all k are 2, 5, 101, 127, 149, 157, 179, 193, 199, 227, 241, 257, 269, 281, 283, 311, 347, 353, 367. In every case the factorization of pkp contains at least one characteristic prime divisor (very different from A306861).

Conjecture: There are an infinite number of -1 terms in this sequence.

Links

Table of n, a(n) for n=1..79.

Formula

If prime(n) is a repunit prime R_k, for some k in A004023 and R_t is the smallest repunit prime such that t > 2*k, then a(n)=R_(t-2*k).

Example

2/2k2, 5/5k5, 7/101k101, 11,13/127k127, 11/149k149, for all k, so a(1)=a(3)=a(26)= a(31)=a(35)=-1. For prime(n)=A004023(2)=R_19, a(n)=R_(317-2*19)=R_279.

Maple

P(p) := proc (p::prime, N::posint := 5000) local n, k, m0, m; n := length(p); for k from 1 to N do m0 := add(10^i, i = 0 .. k-1); m := p*10^(k+n)+m0*10^n+p; if isprime(m) then return k end if; if `mod`(k, 1000) = 0 then print(k) end if end do end proc; P(p) # substitute a prime p here to run the code, it produces an answer (k) if one exists <=N and terms must be computed one at a time.

Crossrefs

Cf. A002275, A004022, A004023, A069687, A306861.

Sequence in context: A329196 A083513 A084277 * A259138 A082710 A240604

Adjacent sequences: A307870 A307871 A307872 * A307874 A307875 A307876

Keyword

sign,base

Author

David James Sycamore, May 02 2019

Status

approved