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%I #41 Sep 09 2016 12:08:30
%S 4,5,7,8,11,12,13,14,15,16,17,18,19,20,21,22,23,24,29,30,32,33,37,38,
%T 39,40,42,44,47,48,53,54,55,56,57,58,59,60,62,63,67,68,71,72,74,75,79,
%U 80,81,82,84,85,89,90,91,92,93,94,97,98,101,102,104,105,106,108,109,110,111,112
%N Minimal increasing sequence beginning with 4 such that n and a(n) are either both prime or both nonprime.
%C For n >= 11, a(n) = A159559(n), which means the two sequences merge.
%C 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.
%C 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?
%C 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
%H Alois P. Heinz, <a href="/A159698/b159698.txt">Table of n, a(n) for n = 1..20000</a>
%H V. Shevelev, <a href="http://arxiv.org/abs/0904.2101">Several results on sequences which are similar to the positive integers</a>, arXiv:0904.2101 [math.NT], 2009.
%F a(1) = 4; for n>1, a(n) = min { m > a(n-1) : m is prime iff n is prime }.
%p a:= proc(n) option remember; local m;
%p if n=1 then 4
%p else for m from a(n-1)+1 while isprime(m) xor isprime(n)
%p do od; m
%p fi
%p end:
%p seq(a(n), n=1..80); # _Alois P. Heinz_, Nov 21 2010
%t 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_ *)
%Y Cf. A159559, A159560, A159615, A159619, A159629, A229019, A229132.
%K nonn,easy
%O 1,1
%A _Vladimir Shevelev_, Apr 20 2009, May 04 2009
%E More terms from _Alois P. Heinz_, Nov 21 2010
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