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A323161 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n<=3) = -n, f(n) = 0 if n-1 is an odd prime, and f(n) = floor((n-1)/2) for all other numbers.

%I A323161 #55 Feb 11 2019 19:22:09
%S A323161 1,2,3,4,5,4,6,4,7,7,8,4,9,4,10,10,11,4,12,4,13,13,14,4,15,15,16,16,
%T A323161 17,4,18,4,19,19,20,20,21,4,22,22,23,4,24,4,25,25,26,4,27,27,28,28,29,
%U A323161 4,30,30,31,31,32,4,33,4,34,34,35,35,36,4,37,37,38,4,39,4,40,40,41,41,42,4,43,43,44,4,45,45,46,46,47,4,48,48,49,49,50,50,51,4,52,52,53,4,54,4
%N A323161 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n<=3) = -n, f(n) = 0 if n-1 is an odd prime, and f(n) = floor((n-1)/2) for all other numbers.
%C A323161 For all i, j: A322809(i) = A322809(j) <=> a(i+1) = a(j+1).
%C A323161 For all i, j: a(i) = a(j) => b(i) = b(j), where b can be, but is not limited to, any of the following sequences: A029834, A049084, A062590, A063377, A064891, A078442 (A049076), A175663, A175682, A269668, A292936, A323162, many of which are related to counting primes in certain kinds of chains or iterations.
%C A323161 Why does this work? Consider the function f given in the definition: based on its properties, we can deduce from the value of f(n) the following information about n:
%C A323161   (A) If f(n) = -2, then n is 2, the only even prime,
%C A323161   (B) If f(n) = -3, then n is 3, the first odd prime,
%C A323161   (C) If f(n) is zero, then n is an even composite preceded by a prime, but we don't know which even composite exactly,
%C A323161   (D) If f(n) > 0 and f(1+2*f(n)) = f(2+2*f(n)), then n is either (D1) an odd composite number, or (D2) an even composite number preceded by an odd composite number, and the said composite number in both cases is 1 + 2*f(n),
%C A323161   (E) If f(n) > 0 and f(1+2*f(n)) <> f(2+2*f(n)), then n is an odd prime > 3, specifically, 1 + 2*f(n).
%C A323161 As this sequence is a restricted growth sequence transform of the said function f, we have a(i) = a(j) <=> f(i) = f(j) for all i, j, thus, even without knowing the value of n, but just a(n), we can find the value of f(n) by searching for the minimal k such that a(k) = a(n), then compute f(k) with that k. Furthermore, any function g defined as g(n) = h(f(n)) [where h is any function], clearly satisfies
%C A323161   a(i) = a(j) => g(i) = g(j), for all i, j. [*]
%C A323161 For instances of such functions g, we can consider many sequences like those sequences b(n) listed above, that have g(n) = 0 for all composite numbers, and g(p) > 0 for all primes p. This is usually the pattern, but there are exceptions, like A323162, which is the characteristic function of A005381, composites n such that n-1 is also composite. These are precisely the numbers that occur twice in this sequence, while all other numbers (including primes), occur just once, that is, reside in their own singular equivalence classes. Thus, it is not guaranteed that all sequences g matching to this sequence (i.e. those satisfying the implication *), even if not false positives in strict sense, would necessarily have some consistent relation to primes, instead, they might contain any random values at the positions given by A093515. However, in the current OEIS, such sequences are exceedingly rare.
%H A323161 Antti Karttunen, <a href="/A323161/b323161.txt">Table of n, a(n) for n = 1..100000</a>
%F A323161 a(1) = 1; for n > 1, a(n) = 1 + A322809(n-1).
%o A323161 (PARI)
%o A323161 up_to = 10000;
%o A323161 rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
%o A323161 A323161aux(n) = if(n<=3,-n,if(isprime(n-1),0,((n-1)>>1))); \\ This implements the function f of the definition.
%o A323161 v323161 = rgs_transform(vector(up_to,n,A323161aux(n)));
%o A323161 A323161(n) = v323161[n];
%Y A323161 Cf. A305801, A305900, A322809.
%Y A323161 Cf. A005381 (numbers that occur twice in this sequence), A093515 (numbers > 1 that occur just once).
%Y A323161 Cf. A010051, A029834, A049076, A049084, A062590, A063377, A064891, A078442, A175663, A175682, A269668, A292936, A323162 (some of the matched sequences).
%K A323161 nonn
%O A323161 1,2
%A A323161 _Antti Karttunen_, Jan 06 2019