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 A254068 Irregular triangle T read by rows in which the entry in row n and column k is given by T(n,k) = 4*A253676(n,k) - 3, k = 1..A253720(n), assuming the 3x+1 (or Collatz) conjecture. 3
 1, 5, 1, 9, 17, 13, 5, 1, 13, 5, 1, 17, 13, 5, 1, 21, 1, 25, 29, 17, 13, 5, 1, 29, 17, 13, 5, 1, 33, 25, 29, 17, 13, 5, 1, 37, 17, 13, 5, 1, 41, 161, 121, 137, 233, 593, 445, 377, 425, 2429, 3077, 577, 433, 325, 61, 53, 5, 1, 45, 17, 13, 5, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Definitions: Let v(y) denote the 2-adic valuation of y. Let N_1 denote the set of odd natural numbers. Let F : N_1 -> N_1 be the map defined by F(x) = (3*x + 1)/2^v(3*x + 1) (cf. A075677). Let F^(k)(x) denote k-fold iteration of F and defined by the recurrence F^(k)(x) = F(F^(k-1)(x)), k>0, with initial condition F^(0)(x) = x. This triangle can be constructed by restricting the initial values to the numbers 4*n - 3, iterating F until 1 is reached (assuming the 3x+1 conjecture) and removing all iterates not congruent to 1 modulo 4. Equivalently, for each n, this is accomplished by iterating (until 1 is reached, assuming the 3x+1 conjecture) the function S defined in A257480 to get the triangle A253676, and finally taking T(n,k) = 4*A253676(n,k) - 3. Conjecture: For each natural number n, there exists a k >= 0, such that F^k(4*n - 3) = 1. Theorem 1: Conjecture 1 is equivalent to the 3x+1 (or Collatz) conjecture. Proof: See A257480. LINKS EXAMPLE T begins: .    1 .    5   1 .    9  17  13   5   1 .   13   5   1 .   17  13   5   1 .   21   1 .   25  29  17  13   5   1 .   29  17  13   5   1 .   33  25  29  17  13   5   1 .   37  17  13   5   1 .   41 161 121 137 233 593 445 377 425 2429 3077 577 433 325 61 53 5 1 .   45  17  13   5   1 .   49  37  17  13   5   1 .   53   5   1 .   57  65  49  37  17  13   5   1} MATHEMATICA v[x_] := IntegerExponent[x, 2]; f[x_] := (3*x + 1)/2^v[3*x + 1]; s[n_] := NestWhileList[(3 + (3/2)^v[1 + f[4*# - 3]]*(1 + f[4*# - 3]))/6 &, n, # > 1 &]; t = Table[4*s[n] - 3, {n, 1, 15}]; Flatten[t] (* Replace Flatten with Grid to display the triangle *) CROSSREFS Cf. A014682, A070165, A075677, A256598, A257480, A253676, A254070. Sequence in context: A100542 A324378 A147404 * A146070 A308504 A040029 Adjacent sequences:  A254065 A254066 A254067 * A254069 A254070 A254071 KEYWORD nonn,tabf AUTHOR L. Edson Jeffery, May 03 2015 STATUS approved

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Last modified September 17 12:00 EDT 2021. Contains 347477 sequences. (Running on oeis4.)