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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. 3
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
For all i, j: A322809(i) = A322809(j) <=> a(i+1) = a(j+1).
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.
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:
(A) If f(n) = -2, then n is 2, the only even prime,
(B) If f(n) = -3, then n is 3, the first odd prime,
(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,
(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),
(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).
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
a(i) = a(j) => g(i) = g(j), for all i, j. [*]
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.
LINKS
FORMULA
a(1) = 1; for n > 1, a(n) = 1 + A322809(n-1).
PROG
(PARI)
up_to = 10000;
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; };
A323161aux(n) = if(n<=3, -n, if(isprime(n-1), 0, ((n-1)>>1))); \\ This implements the function f of the definition.
v323161 = rgs_transform(vector(up_to, n, A323161aux(n)));
A323161(n) = v323161[n];
CROSSREFS
Cf. A005381 (numbers that occur twice in this sequence), A093515 (numbers > 1 that occur just once).
Cf. A010051, A029834, A049076, A049084, A062590, A063377, A064891, A078442, A175663, A175682, A269668, A292936, A323162 (some of the matched sequences).
Sequence in context: A244904 A338885 A238288 * A152739 A279614 A212639
KEYWORD
nonn
AUTHOR
Antti Karttunen, Jan 06 2019
STATUS
approved

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Last modified March 28 12:59 EDT 2024. Contains 371254 sequences. (Running on oeis4.)