A115774 -id:A115774 - cp's OEIS frontend

A115775 Sequence A115774 in binary.

0, 101, 1010, 10100, 10101, 101000, 101010, 1010000, 1010100, 1010101, 10100000, 10101000, 10101010, 101000000, 101010000, 101010100, 101010101, 1010000000, 1010000101, 1010100000, 1010101000, 1010101010, 10100000000

Offset: 0

Antti Karttunen, Jan 30 2006

Cf. a(n) = A007088(A115774(n)).

A062052 Numbers with exactly 2 odd integers in their Collatz (or 3x+1) trajectory.

Original entry on oeis.org

5, 10, 20, 21, 40, 42, 80, 84, 85, 160, 168, 170, 320, 336, 340, 341, 640, 672, 680, 682, 1280, 1344, 1360, 1364, 1365, 2560, 2688, 2720, 2728, 2730, 5120, 5376, 5440, 5456, 5460, 5461, 10240, 10752, 10880, 10912, 10920, 10922, 20480, 21504, 21760, 21824

Offset: 1

David W. Wilson

nonn

The Collatz (or 3x+1) function is f(x) = x/2 if x is even, 3x+1 if x is odd.
The Collatz trajectory of n is obtained by applying f repeatedly to n until 1 is reached.
The sequence consists of terms of A002450 and their 2^k multiples. The first odd integer in the trajectory is one of the terms of A002450 and the second odd one is the terminal 1. - Antti Karttunen, Feb 21 2006
This sequence looks to appear first in the literature on page 1285 in R. E. Crandall.

			The Collatz trajectory of 5 is (5,16,8,4,2,1), which contains 2 odd integers.

Reinhard Zumkeller and T. D. Noe, Table of n, a(n) for n = 1..1000 (first 100 terms from Reinhard Zumkeller)
R. E. Crandall, On the 3x+1 problem, Math. Comp., 32 (1978) 1281-1292.
J. Shallit and D. Wilson, The "3x+1" Problem and Finite Automata, Bulletin of the EATCS #46 (1992) pp. 182-185.
Eric Weisstein's World of Mathematics, Collatz Problem
Wikipedia, Collatz conjecture
Index entries for sequences related to 3x+1 (or Collatz) problem
Index entries for 2-automatic sequences.

Cf. A062053, A062054, A062055, A062056, A062057, A062058, A062059, A062060, A122196, A122197.
Is this a subset of A115774?
Column k=2 of A354236.

import Data.List (elemIndices)
a062052 n = a062052_list !! (n-1)
a062052_list = map (+ 1) $ elemIndices 2 a078719_list
-- Reinhard Zumkeller, Oct 08 2011

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; countOdd[lst_] := Length[Select[lst, OddQ]]; Select[Range[22000], countOdd[Collatz[#]] == 2 &] (* T. D. Noe, Dec 03 2012 *)

for(n=2,100000,s=n; t=0; while(s!=1,if(s%2==0,s=s/2,s=3*s+1; t++); if(s*t==1,print1(n,","); ); ))

def a(n):
    l=[n, ]
    while True:
        if n%2==0: n//=2
        else: n = 3*n + 1
        if n not in l:
            l.append(n)
            if n<2: break
        else: break
    return len([i for i in l if i % 2])
print([n for n in range(1, 22001) if a(n)==2]) # Indranil Ghosh, Apr 14 2017

A078719(a(n)) = 2; A006667(a(n)) = 1.

a(n) = 2^x * (4^y - 1)/3 where x = A122196(n) - 1 and y = A122197(n) + 1. - Alan Michael Gómez Calderón, Jan 16 2025 after Antti Karttunen

A115772 Integers i such that 13*i = A048720bi(21,i).

Original entry on oeis.org

0, 5, 10, 15, 20, 21, 30, 31, 40, 42, 45, 47, 60, 61, 62, 63, 80, 84, 85, 90, 94, 95, 120, 122, 124, 125, 126, 127, 160, 165, 168, 170, 173, 175, 180, 181, 188, 189, 190, 191, 240, 244, 245, 248, 250, 252, 253, 254, 255, 320, 330, 336, 340, 341, 346, 350, 351

Offset: 0

Antti Karttunen, Jan 30 2006

nonn

Here * stands for ordinary multiplication and A048720 is the carryless (GF(2)[X]) multiplication.

a(n) appears to be the set of all n that can be expressed as x OR 4x for the bitwise OR operation. [From Gary Detlefs Dec 20 2010]

Index entries for sequences defined by congruent products between domains N and GF(2)[X]

Row 13 of A115872. Cf. A048717, A115767, A115770. Superset of A115774 ? A115776 gives the terms which are not in A115774. A115773 shows this sequence in binary.

Cf. A178891

A115770 Integers i such that 7*i = A048720bi(11, i), where A048720bi implements the dyadic function given in A048720 (see A001317).

Original entry on oeis.org

0, 7, 14, 15, 28, 30, 31, 56, 60, 62, 63, 112, 120, 124, 126, 127, 224, 240, 248, 252, 254, 255, 448, 455, 480, 496, 504, 508, 510, 511, 896, 903, 910, 911, 960, 967, 992, 1008, 1016, 1020, 1022, 1023, 1792, 1799, 1806, 1807, 1820, 1822, 1823, 1920

Offset: 0

Antti Karttunen, Jan 30 2006

nonn

Here * stands for ordinary multiplication and A048720 is the carryless (GF(2)[X]) multiplication.

Index entries for sequences defined by congruent products between domains N and GF(2)[X]

Row 7 of A115872 (conjecture: also row 5).
Cf. A001317, A048717, A048720, A115767, A115772, A115774, A115776.
A115771 shows this sequence in binary.

A115776 Integers i such that 13i = A048720bi(21,i), but 15i <> A048720bi(23,i).

Original entry on oeis.org

15, 30, 31, 45, 47, 60, 61, 62, 63, 90, 94, 95, 120, 122, 124, 125, 126, 127, 165, 173, 175, 180, 181, 188, 189, 190, 191, 240, 244, 245, 248, 250, 252, 253, 254, 255, 330, 346, 350, 351, 360, 362, 376, 378, 380, 381, 382, 383, 480, 488, 490, 496, 500

Offset: 1

Antti Karttunen, Jan 30 2006

nonn

Here * stands for ordinary multiplication and A048720 is the carryless (GF(2)[X]) multiplication.

Index entries for sequences operating on GF(2)[X]-polynomials

Cf. Setwise difference of A115772 and A115774. A115781 shows this sequence in binary.

cp's OEIS Frontend

A115775 Sequence A115774 in binary.

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A062052 Numbers with exactly 2 odd integers in their Collatz (or 3x+1) trajectory.

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A115772 Integers i such that 13*i = A048720bi(21,i).

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A115770 Integers i such that 7*i = A048720bi(11, i), where A048720bi implements the dyadic function given in A048720 (see A001317).

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A115776 Integers i such that 13*i = A048720bi(21,i), but 15*i <> A048720bi(23,i).

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A115776 Integers i such that 13i = A048720bi(21,i), but 15i <> A048720bi(23,i).