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If a number c satisfies the Collatz conjecture, this means that there exists k >= log_2(c) such that (1) c <= 2^k and each descendant c_i of c verifies satisfies c_i <= 2^(k+i), i -> {1,2,3,...,oo}, and (2) every ancestor c_j of c verifies satisfies c_j <= 2^(k-j), j -> {1,...,k}. Then a(c) = 2^k (see second PARI PROG).

Now, if c does NOT satisfy the Collatz conjecture, this means for EVERY k >= log_2(c) (as k tends to infinity) (1) c <= 2^k and each descendant c_i of c verifies satisfies c_i <= 2^(k+i), i -> {1,2,3,...,oo} but there is a j-th ancestor of c, the number c_j, that is c_j > 2^(k-j), j -> {1,...,k}. Then we can say arbitrarily that a(c) = -1 or a(c) = οο.

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2) every ancestor c_j of c verifies c_j <= 2^(k-j), j -> {1,..,k}. Then a(c) = 2^k (see second PARI PROG).

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(PARI) for(k = 1, 37, n = k; s = 0; while(n > 1, if(n%2 = = 0, n/=2; s++, n = 3*n + 1; s++)); print1(2^s", "))

(PARI) a(nn) = kk = 2^floor(log(nn) / log(2) + .5); n = nn; k = kk; while(n < k, if(n%2 == 0, n/=2, n = 3*n + 1); k/=2; if(n > k, kk*=2; k = kk; n = nn)); print1(kk", ")

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As we see in the example the numbers 7, 2 ^ 16 = 65536 = 2 ^ 16 belong to the 16th generation of Collatz numbers, starting from 1. This means that they have a finite number of ancestors 16 and will leave an infinite number of descendants. I find it difficult to understand what would be the finite number c that would not belong to any generation, would leave infinite descendants and would come from infinite ancestors, given that the members of each previous generation are dwindling.

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As we see in the example the numbers 7, 2 ^ 16 = 65536 belong to the 16th generation of Collatz numbers, starting from 1. This means that they have a finite number of ancestors 16 and will leave an infinite number of descendants. I find it difficult to understand what would be the finite number c that would not belong to any generation, would leave infinite descendants and would come from infinite ancestors, given that the members of each previous generation are dwindling.