A078350 -id:A078350 - cp's OEIS frontend

A359131 Number of odd primes in the Collatz trajectory of A177000(n).

0, 2, 1, 5, 4, 2, 3, 6, 5, 6, 2, 10, 8, 9, 7, 8, 6, 11, 9, 4, 7, 5, 8, 3, 9, 7, 4, 2, 3, 6, 9, 7, 10, 5, 8, 6, 11, 4, 2, 10, 3, 9, 10, 11, 12, 3, 6, 5, 11, 4, 10, 5, 3, 6, 7, 9, 7, 2, 7, 5, 8, 6, 4, 7, 5, 8, 3, 6, 9, 10, 5, 8, 3, 7, 8, 8, 13, 11, 11, 9, 12, 2

Offset: 1

N. J. A. Sloane, Dec 29 2022, following a suggestion from Rudolfo Nieves

nonn

Equivalently, a(n) = A055509(A177000(n)).

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A055509, A078350, A078373, A177000.

a(38) and beyond from Michael S. Branicky, Dec 30 2022

A319227 a(n) is the number of twin primes in the Collatz trajectory of n.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 2, 0, 2, 0, 1, 1, 0, 2, 0, 0, 0, 2, 2, 0, 0, 1, 0, 1, 2, 0, 0, 2, 1, 0, 0, 0, 2, 0, 0, 2, 2, 2, 2, 0, 0, 0, 2, 1, 0, 0, 0, 1, 2, 2, 1, 0, 0, 0, 0, 2, 2, 1, 2, 0, 0, 0, 0, 0, 2, 2, 2, 0, 0, 0, 0, 2, 1, 2, 0, 2, 1, 2, 2, 0, 0, 0, 0, 0, 0, 2, 2

Offset: 1

Michel Lagneau, Sep 14 2018

nonn

Conjecture: a(n) <=2.
For a(n) = 2, the corresponding twin primes are (5, 7) and (11, 13) or (11, 13) and (17, 19).
This sequence is generalizable: let a(n, p, p+2q) be the number of pairs of primes of form (p, p+2q) in the Collatz trajectory of n, q = 1, 2,... It is conjectured that a(n, p, p+2q) < =2. (see the table below).
+----------------+---------------------------------+
| pairs of prime |     pairs of prime numbers      |
|     numbers    |    in the Collatz trajectory    |
|                |    when a(n, p, p+2q) = 2       |
+----------------+---------------------------------+
|    (p, p+2)    |     (5, 7) and (11, 13)         |
|                |  or (11, 13) and (17, 19)       |
+----------------+---------------------------------+
|    (p, p+4)    |     (7, 11) and (13, 17)        |
+----------------+---------------------------------+
|    (p, p+6)    |     (41, 47) and (47, 53)       |
|                |  or (47, 53) and (97, 103)      |
|                |  or (47, 53) and (587, 593)     |
+----------------+---------------------------------+
|    (p, p+8)    |     (23, 31) and (53, 61)       |
+----------------+---------------------------------+
|    (p, p+10)   |     (61, 71) and (73, 83)       |
|                |  or (61, 71) and (283, 293)     |
|                |  or (61, 71) and (577, 587)     |
+----------------+---------------------------------+
|    (p, p+12)   |     (71, 83) and (251, 263)     |
|                |  or (251, 263) and (467, 479)   |
|                |  or (251, 263) and (479, 491)   |
|                |  or (251, 263) and (1607, 1619) |
+----------------+---------------------------------+
|    (p, p+14)   |     No results for n <= 10^6    |
+----------------+---------------------------------+
...................................................

			a(7) = 2 because the Collatz trajectory of 7 is 7 -> 22 -> 11 -> 34 -> 17 -> 52 -> 26 -> 13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1 with two twin primes: (5, 7) and (11, 13).

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

Cf. A006370, A070165, A078350, A196871, A280408.

nn:=10^8:
for n from 1 to 100 do:
   m:=n:lst:={}:
      for i from 1 to nn while(m<>1) do:
        if irem(m, 2)=0
         then
         m:=m/2:
         else
         lst:=lst union {m}:m:=3*m+1:
       fi:
     od:
     n0:=nops(lst):it:=0:
     for j from 1 to n0-1 do:
     if isprime(lst[j]) and isprime(lst[j+1]) and lst[j+1]=lst[j]+2
     then it:=it+1:else fi:
      od:
    printf(`%d, `,it):
    od:

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A359131 Number of odd primes in the Collatz trajectory of A177000(n).

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A362958 a(n) is the number of primes in a Collatz orbit started at A078373(n).

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A319227 a(n) is the number of twin primes in the Collatz trajectory of n.

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