A218344 Smallest k such that k*(n-th composite)+1 is prime.

1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 3, 4, 2, 4, 1, 1, 3, 2, 3, 2, 1, 5, 2, 1, 1, 2, 4, 1, 2, 4, 2, 2, 1, 2, 6, 2, 4, 1, 1, 5, 2, 3, 2, 1, 2, 2, 1, 1, 2, 2, 3, 6, 1, 3, 2, 1, 4, 12, 2, 4, 1, 2, 6, 3, 4, 3, 2, 1, 2, 2, 1, 1, 3, 2, 1, 1, 3, 2, 1, 2, 4, 2, 8, 6, 2

Offset

1,3

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Example

The composite numbers are A002808.  a(1) is 1 since the first composite number is 4, and 4*1+1=5, a prime.  a(14)=3 since the 14th composite is 24, and 24*3+1=73 prime, while 25 and 49 are not.

Mathematica

t={}; For[k = 4, k < 200, k++, If[!PrimeQ[k], Mult = 1; While[! PrimeQ[k*Mult + 1], Mult = Mult + 1]; AppendTo[t, Mult]]]; t
sk[n_]:=Module[{k=1}, While[!PrimeQ[k*n+1], k++]; k]; nn=150; With[{cmps= Complement[ Range[4, nn], Prime[Range[PrimePi[nn]]]]}, sk/@cmps] (* Harvey P. Dale, Apr 16 2013 *)

Prog

(Python)
from sympy import composite, isprime
def a(n):
    cn, k = composite(n), 1
    while not isprime(k*cn + 1): k += 1
    return k
print([a(n) for n in range(1, 89)]) # Michael S. Branicky, Jun 07 2022

Crossrefs

Cf. A002808.

Sequence in context: A211263 A303827 A323116 * A211272 A298600 A292470

Adjacent sequences: A218341 A218342 A218343 * A218345 A218346 A218347

Keyword

nonn,easy

Author

William J. Keith, Oct 26 2012

Status

approved