A372443 -id:A372443 - cp's OEIS frontend

A372448 a(n) is the 2-adic valuation of 1 + 3*{the n-th iterate of 27 with A371094}.

1, 4, 9, 19, 39, 79, 160, 322, 645, 1292, 2585, 5171, 10344, 20689, 41379, 82759, 165520, 331043, 662087, 1324175, 2648352, 5296705, 10593412, 21186825, 42373651, 84747303, 169494607, 338989215, 677978433, 1355956867, 2711913735, 5423827471, 10847654946, 21695309894, 43390619790, 86781239584, 173562479171, 347124958343

Offset: 0

Antti Karttunen, May 04 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 0..1001

Cf. A371093, A371094, A372443, A372444, A372447, A372449.

A000265(n) = (n>>valuation(n,2));
A372443(n) = { my(x=27); while(n, x=A000265(3*x+1); n--); (x); };
A371093(n) = valuation(1+3*n,2);
A372448(n) = if(!n,1,2*A372448(n-1)+A371093(A372443(n)));

a(n) = A371093(A372444(n)).

a(0) = 1, and for n > 0, a(n) = 2*a(n-1) + A371093(A372443(n)).

A372362 a(n) is the 2-adic valuation of 1 + 3*{the n-th iterate of 27 with Reduced Collatz-function R}.

Original entry on oeis.org

1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 4, 2, 2, 4, 3, 1, 1, 5, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 0

Antti Karttunen, May 06 2024

nonn

Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A371093, A372443, A372447, A372448.

The first 41 terms form the row 13 of A351122.

A000265(n) = (n>>valuation(n,2));
A371093(n) = valuation(1+3*n,2);
A372443(n) = { my(x=27); while(n, x=A000265(3*x+1); n--); (x); };
A372362(n) = A371093(A372443(n));

a(n) = A371093(A372443(n)).

A372451 a(n) = A372449(1+n) - A372449(n).

Original entry on oeis.org

3, 5, 11, 21, 40, 81, 161, 324, 647, 1293, 2587, 5172, 10346, 20691, 41380, 82761, 165521, 331045, 662088, 1324177, 2648354, 5296706, 10593414, 21186826, 42373653, 84747305, 169494608, 338989217, 677978434, 1355956869, 2711913737, 5423827472, 10847654948, 21695309895, 43390619792, 86781239585, 173562479173, 347124958345

Offset: 0

Antti Karttunen, May 05 2024

nonn

a(n) tells how many bits the length of the binary expansion grows when we go from A372444(n) to A372444(1+n).

Antti Karttunen, Table of n, a(n) for n = 0..1001

First differences of A372449.
Column 14 of A372356.
Cf. A000523, A372443, A372444.

A372451(n) = (A372449(1+n)-A372449(n)); \\ Needs also code from A372449.

a(n) = A000523(A372444(1+n)) - A000523(A372444(n)).

A372450 a(n) = k, if A086893(k) is the first term of A086893 reached on the trajectory of reduced Collatz-function R, when starting from 2n-1, or -1 if no term of A086893 is ever encountered.

Original entry on oeis.org

1, 2, 3, 4, 4, 4, 4, 6, 4, 4, 5, 6, 4, 6, 4, 6, 4, 6, 4, 4, 6, 4, 4, 6, 4, 4, 6, 6, 4, 4, 6, 6, 4, 4, 4, 6, 6, 7, 4, 4, 6, 6, 7, 4, 4, 6, 6, 6, 6, 4, 4, 6, 4, 6, 6, 6, 7, 4, 4, 4, 6, 4, 6, 4, 6, 4, 4, 4, 6, 4, 6, 6, 6, 6, 4, 9, 4, 6, 4, 6, 6, 6, 6, 6, 4, 6, 4, 6, 4, 4, 4, 6, 4, 4, 6, 4, 6, 6, 4, 6, 9, 4, 4, 6, 4, 4, 8, 6

Offset: 1

Antti Karttunen, May 03 2024

nonn

The length of the binary expansion of the first term of A086893 that comes along when starting from x = 2*n-1 and then repeating the operation x -> A000265(3*x+1). If 2n-1 itself is in A086893, then its binary length is used.

Terms A016789(n) = 2, 5, 8, 11, 14, 17, ... occur only once in this sequence because A086893(A016789(n)) are all multiples of 3: 3, 21, 213, 1365, 13653, 87381, 873813, 5592405, 55924053, 357913941, ..., while the terms of A075677 never are. Note that all terms > 1 of A086893 are just one or two invocations of R away from 1.

			a(11) = 5 because the first term of A086893 that occurs on the trajectory of 21 (= 2*11-1) is 21 = A086893(5).
a(14) = 6 because the first term of A086893 that occurs on the trajectory of 27 (= 2*14-1) is A372443(39) = 53 = A086893(6).

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A000265, A000523, A016789, A075677, A086893, A372358, A372443.

A000265(n) = (n>>valuation(n,2));
A000523(n) = logint(n,2);
A086893(n) = (if(n%2, 2^(n+1), 2^(n+1)+2^(n-1))\3);
A372358(n) = bitxor(n, A086893(1+A000523(n)));
A372450(n) = { n=2*n-1; while(A372358(n)>0, n=A000265(3*n+1)); (1+A000523(n)); };

cp's OEIS Frontend

A372448 a(n) is the 2-adic valuation of 1 + 3*{the n-th iterate of 27 with A371094}.

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A372362 a(n) is the 2-adic valuation of 1 + 3*{the n-th iterate of 27 with Reduced Collatz-function R}.

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A372451 a(n) = A372449(1+n) - A372449(n).

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A372450 a(n) = k, if A086893(k) is the first term of A086893 reached on the trajectory of reduced Collatz-function R, when starting from 2n-1, or -1 if no term of A086893 is ever encountered.

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