A159698 Minimal increasing sequence beginning with 4 such that n and a(n) are either both prime or both nonprime.

4, 5, 7, 8, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 29, 30, 32, 33, 37, 38, 39, 40, 42, 44, 47, 48, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 67, 68, 71, 72, 74, 75, 79, 80, 81, 82, 84, 85, 89, 90, 91, 92, 93, 94, 97, 98, 101, 102, 104, 105, 106, 108, 109, 110, 111, 112

Offset: 1

Author

Vladimir Shevelev, Apr 20 2009, May 04 2009

Keywords

• nonn
• easy

Comments

For n >= 11, a(n) = A159559(n), which means the two sequences merge.

We may define other sequences a(p-1,n), p prime, which start a(p-1,1)=p-1 and with the same property of n and a(p-1,n) being jointly prime or nonprime.

We find that for p=7, 11 and 13, the sequences a(6,n), a(10,n) and a(12,n) also merge with the current sequence for sufficiently large n. Does this also hold for primes >=17?

It was verified for primes p with 7<=p<=223 that this sequence a(4,n) and a(p-1,n) eventually merge. The corresponding values of n are 47, 683, 1117, 6257, 390703. - Alois P. Heinz, Mar 09 2011

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..20000
• V. Shevelev, Several results on sequences which are similar to the positive integers, arXiv:0904.2101 [math.NT], 2009.

Crossrefs

Cf. A159559, A159560, A159615, A159619, A159629, A229019, A229132.

Programs

Maple

a:= proc(n) option remember; local m;
      if n=1 then 4
    else for m from a(n-1)+1 while isprime(m) xor isprime(n)
         do od; m
      fi
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, Nov 21 2010

Mathematica

a[n_] := a[n] = If[n==1, 4, For[m = a[n-1]+1, Xor[PrimeQ[m], PrimeQ[n]], m++]; m]; Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Jan 31 2016, after Alois P. Heinz *)

Formula

a(1) = 4; for n>1, a(n) = min { m > a(n-1) : m is prime iff n is prime }.

Extensions

More terms from Alois P. Heinz, Nov 21 2010