Alessandro Polcini - cp's OEIS frontend

A320028 a(n) is the first prime encountered when running the Collatz algorithm (halving and tripling steps) on the number n.

2, 3, 2, 5, 3, 7, 2, 7, 5, 11, 3, 13, 7, 23, 2, 17, 7, 19, 5, 2, 11, 23, 3, 19, 13, 41, 7, 29, 23, 31, 2, 19, 17, 53, 7, 37, 19, 59, 5, 41, 2, 43, 11, 17, 23, 47, 3, 37, 19, 29, 13, 53, 41, 83, 7, 43, 29, 59, 23, 61, 31, 137, 2, 37, 19, 67, 17, 13, 53, 71, 7, 73, 37, 113, 19, 29, 59, 79, 5, 61, 41, 83, 2, 2, 43, 131

Offset: 2

Alessandro Polcini, Oct 03 2018

nonn

A modified version of the halving and tripling Collatz algorithm, which stops as soon as the starting number becomes a prime (instead of stopping when the starting number reaches 1).

The plot of this sequence "completes" or "fills" the lower (empty) part of plot of A270570 and evolves in a similar fashion.

			a(4) is 2 because 4/2 = 2 and 2 is prime.
a(6) is 3 because 6/2 = 3 and 3 is prime.
a(15) is 23 because 15*3 + 1 = 46; 46/2 = 23 and 23 is prime.
a(18) is 7 because 18/2 = 9; 9*3 + 1 = 28; 28/2 = 14; 14/2 = 7 and 7 is prime.

Alessandro Polcini, Table of n, a(n) for n = 2..10000 (a(2024) corrected by Michel Marcus, Jun 14 2022)
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A087272, A270570.

int collatzPrime(int i) {
    while(!BigInteger.valueOf(i).isProbablePrime(10) && i > 1) {
        if(i % 2 == 0)
            i /= 2;
        else
            i = 3 * i + 1;
    }
    return i;
}

Array[NestWhile[If[EvenQ@ #, #/2, 3 # + 1] &, #, ! PrimeQ@ # &] &, 86, 2] (* Michael De Vlieger, Nov 07 2018 *)

a(n) = {while (!isprime(n), if (n % 2, n = 3*n+1, n = n/2);); n;} \\ Michel Marcus, Oct 28 2018

a(n) <= A087272(n). - Rémy Sigrist, Oct 08 2018

A319936 Numbers with more than one Collatz tripling step whose odd Collatz trajectory does not contain primes.

Original entry on oeis.org

113, 227, 453, 906, 909, 1812, 1813, 1818, 2417, 3624, 3626, 3636, 3637, 7248, 7252, 7253, 7272, 7281, 9669, 14496, 14504, 14544, 14549, 14562, 14563, 19338, 28992, 29008, 29013, 29088, 29124, 29125, 30559, 38676, 38677, 38833, 38835, 45839, 54327, 57984

Offset: 1

Alessandro Polcini, Oct 10 2018

nonn

The starting number itself is not counted in the trajectory, otherwise prime numbers like 113 or 227 wouldn't appear in this sequence.

The "odd Collatz trajectory" of a number k is the subset of odd numbers of the full Collatz trajectory of k.

			113 is in this sequence because 113*3+1 = 340; 340/2 = 170; 170/2 = 85; 85*3+1 = 256, which goes to 1. The trajectory has 2 (> 1) tripling steps and 85 isn't a prime.
114 is not in this sequence because 114/2 = 57; 57*3+1 = 172; 172/2 = 86; 86/2 = 43, which is a prime, and this trajectory has more than 1 tripling step.

Cf. A002450, A006577.

for(int i = 0; i < DIM; i++) {
    if(!collatzAtLeastOnePrime(c) && collatzTriplingSteps(c) > 1)
        System.out.print(c + ", ");
}
boolean collatzAtLeastOnePrime(int i) {
    //first step outside the while loop...
  if(i % 2 == 0)
        i /= 2;
    else
        i = 3 * i + 1;
  //...otherwise prime numbers like 113 or 227 would be excluded
    while(i > 1) {
        if(i % 2 == 0) {
            i /= 2;
        }
        else {
            if(BigInteger.valueOf(i).isProbablePrime(10))
                return true;
            i = 3 * i + 1;
        }
    }
    return false;
}

Select[Range[3, 60000], And[Count[#, ?OddQ] > 1, NoneTrue[Rest@ #, PrimeQ]] &@ NestWhileList[If[EvenQ@ #, #/2, 3 # + 1] &, #, # > 2 &, 1, Infinity, -1] &] (* _Michael De Vlieger, Nov 07 2018 *)

A216189 Numbers n whose odd Collatz steps (except for 1) are all primes.

Original entry on oeis.org

3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 17, 19, 20, 21, 22, 24, 25, 26, 28, 29, 34, 35, 37, 38, 39, 40, 44, 45, 48, 49, 52, 53, 56, 58, 59, 67, 68, 69, 74, 76, 77, 80, 85, 87, 88, 89, 96, 99, 101, 104, 106, 112, 116, 117, 118, 119, 131, 134, 136, 141, 148, 149

Offset: 1

Alessandro Polcini, Oct 10 2018

nonn

Powers of 2 are not included as they don't have odd Collatz steps. The number n itself (if odd) is not counted as an odd step. [Corrected by Jianing Song, Dec 09 2018]

			7 is in this sequence because 7*3+1 = 22; 22/2 = [11]; 11*3+1 = 34; 34/2 = [17]; 17*3+1 = 52; 52/2 = 26; 26/2 = [13]; 13*3+1 = 40; 40/2 = 20; 20/2 = 10; 10/2 = [5]; 5*3+1 = 16; 16/2 = 8; 8/2 = 4; 4/2 = 2; 2/2 = 1. 11, 17, 13 and 5 are all primes.
15 is not in this sequence because 15*3+1 = 46; 46/2 = [23]; 23*3+1 = 70; 70/2 = 35, which isn't a prime number.

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

Cf. A006577, A177000.

Select[Range[3, 150], And[AllTrue[Select[Rest@ #2, OddQ], PrimeQ], !IntegerQ@ Log2@ #1] & @@ {#, NestWhileList[If[EvenQ@ #, #/2, 3 # + 1] &, #, # > 2 &, 1, Infinity, -1]} &] (* Michael De Vlieger, Nov 07 2018 *)

A320020 Numbers whose Collatz trajectory always alternates between a halving and a tripling step until a power of 2 is reached.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 16, 21, 32, 42, 64, 85, 128, 151, 170, 227, 256, 302, 341, 454, 512, 682, 1024, 1365, 2048, 2730, 4096, 5461, 8192, 10922, 14563, 16384, 21845, 29126, 32768, 43690, 65536, 87381, 131072, 174762, 262144, 349525, 524288, 699050, 932067

Offset: 1

Alessandro Polcini, Oct 03 2018

nonn

Any number whose Collatz trajectory has two or more consecutive halving steps (before reaching a power of 2) is not included in this sequence.
The b-file was generated using the reverse Collatz algorithm.
Instead of going: n -> 3n + 1 (if odd), n/2 (if even), this algorithm starts from a power of 2 and goes: n -> (n - 1)/3 (if possible) and n*2, repeating these two consecutive steps as long as the number n is divisible by 3. The results are progressively added to a dynamic array, which gets sorted by ascending order once the algorithm terminates.
Example 1: 8 -> (8 - 1) and the calculation stops because 7 isn't divisible by 3. No numbers are added to the dynamic array.
Example 2: 16 -> (16 - 1)/3 = 5; 5*2 = 10; (10 - 1)/3 = 3; 3*2 = 6; (6 - 1) and the calculation stops because 5 isn't divisible by 3. The numbers 5, 10, 3, 6 are then added to the dynamic array.
Example 3: 64 -> (64 - 1)/3 = 21; 21*2 = 42; (42 - 1) and the calculation stops because 41 isn't divisible by 3. The numbers 21 and 42 are then added to the dynamic array.

			6 is in the sequence because 6/2 = 3 (halving step); 3*3 + 1 = 10 (tripling step); 10/2 = 5 (halving step); 5*3 + 1 = 16 (tripling step) and a power of 2 is reached.
7 is not in this sequence because 7*3 + 1 = 22 (tripling step); 22/2 = 11 (halving step); 11*3 + 1 = 34 (tripling step); 34/2 = 17 (halving step); 17*3 + 1 = 52 (tripling step); 52/2 = 26 (halving step); 26/2 = 13 (another halving step).

Alessandro Polcini, Table of n, a(n) for n = 1..1000
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006577.

for(BigInteger pow = TWO; pow.compareTo(END) < 0; pow = pow.multiply(TWO)) {
    terms.add(pow); //otherwise numbers like 8, 32, 128, etc. aren't added
    BigInteger newPow = pow.subtract(ONE);
    while(newPow.mod(THREE).compareTo(ZERO) == 0) {
        terms.add(newPow.add(ONE));
        newPow = newPow.divide(THREE);
        terms.add(newPow);
        newPow = newPow.add(newPow);
        terms.add(newPow);
        newPow = newPow.subtract(ONE);
    }
}
terms.sort(Comparator.naturalOrder()); //sorting by ascending order
for(int i = 1; i < terms.size(); i++) {
    if(terms.get(i).compareTo(terms.get(i - 1)) == 0) {
        terms.remove(i); //to remove duplicates
    }
}

isp2(n) = (n==1) || (n==2) || (ispower(n,,&p) && (p==2));
isok(n) = {while (! isp2(n), if (n % 2, newn = (3*n+1), newn = n/2); if (((n % 2) == (newn % 2)), return (0)); n = newn;); return (1);} \\ Michel Marcus, Oct 07 2018

A298509 First differences of A023108.

Original entry on oeis.org

99, 99, 99, 99, 97, 2, 97, 2, 89, 8, 91, 8, 509, 2, 88, 2, 88, 2, 88, 2, 88, 2, 88, 2, 50, 497, 2, 88, 2, 88, 2, 88, 2, 88, 2, 88, 2, 50, 497, 2, 88, 2, 88, 2, 88, 2, 88, 2, 88, 2

Offset: 1

Alessandro Polcini, Jan 20 2018

It appears that this sequence has a very regular pattern, as shown by the attached .png graphical visualization.

Alessandro Polcini, Table of n, a(n) for n = 1..100000
Alessandro Polcini, Lychrel steps, graphically visualized in a 1000x1000 grid

Cf. A023108.

A282671 Twice composite numbers.

Original entry on oeis.org

8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 44, 48, 50, 52, 54, 56, 60, 64, 66, 68, 70, 72, 76, 78, 80, 84, 88, 90, 92, 96, 98, 100, 102, 104, 108, 110, 112, 114, 116, 120, 124, 126, 128, 130, 132, 136, 138, 140, 144, 148, 150, 152, 154, 156, 160, 162, 164, 168, 170, 172, 174

Offset: 1

Alessandro Polcini, Feb 20 2017

Even numbers greater than 2 that do not appear in A001747.

Cf. A001747, A139270, A176100.

is(n)=n%2==0 && n>6 && !isprime(n/2) \\ Charles R Greathouse IV, Feb 21 2017

a(n) = 2*A002808(n). - R. J. Mathar, Feb 23 2017

a(n) = A139270(n+1). - R. J. Mathar, Feb 25 2017

A274489 a(n) = floor(sinh(n) / n^2).

Original entry on oeis.org

1, 0, 1, 1, 2, 5, 11, 23, 50, 110, 247, 565, 1308, 3067, 7264, 17355, 41790, 101327, 247205, 606456, 1495255, 3703422, 9210589, 22994029, 57603919, 144770421, 364916488, 922357821, 2337297441

Offset: 1

Alessandro Polcini, Nov 10 2016

This sequence has some elements in common with A018112 (-> elements 1, 2, 5, 11, 23, 50, 110).

The sum of the reciprocals of a(n) (for n>=2) converges to 2.870623225848376377857...

Table[Floor[Sinh[n]/n^2],{n,30}] (* Harvey P. Dale, Nov 15 2018 *)

a(n)=localprec(19); localprec((n-log(2))\log(10)+9); sinh(n)\n^2 \\ Charles R Greathouse IV, Nov 10 2016

a(n) = floor(sinh(n)/(n^2)), for n >= 1.

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User: Alessandro Polcini

A320028 a(n) is the first prime encountered when running the Collatz algorithm (halving and tripling steps) on the number n.

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A319936 Numbers with more than one Collatz tripling step whose odd Collatz trajectory does not contain primes.

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A216189 Numbers n whose odd Collatz steps (except for 1) are all primes.

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A320020 Numbers whose Collatz trajectory always alternates between a halving and a tripling step until a power of 2 is reached.

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A298509 First differences of A023108.

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A282671 Twice composite numbers.

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A274489 a(n) = floor(sinh(n) / n^2).

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