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

1,1

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.

Formula

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

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 *)

Crossrefs

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

Sequence in context: A293856 A098416 A005556 * A288931 A191276 A228919

Adjacent sequences: A159695 A159696 A159697 * A159699 A159700 A159701

Keyword

nonn,easy

Author

Vladimir Shevelev, Apr 20 2009, May 04 2009

Extensions

More terms from Alois P. Heinz, Nov 21 2010

Status

approved