

A159698


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


8



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
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OFFSET

1,1


COMMENTS

For n >= 11, a(n) = A159559(n), which means the two sequences merge.
We may define other sequences a(p1,n), p prime, which start a(p1,1)=p1 and with the same property of n and a(p1,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(p1,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(n1) : 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(n1)+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[n1]+1, Xor[PrimeQ[m], PrimeQ[n]], m++]; m]; Table[a[n], {n, 1, 80}] (* JeanFrançois Alcover, Jan 31 2016, after Alois P. Heinz *)


CROSSREFS

Cf. A159559, A159560, A159615, A159619, A159629, A229019, A229132.
Sequence in context: A032722 A098416 A005556 * A191276 A228919 A047377
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



