A214850 -id:A214850 - cp's OEIS frontend

A224504 a(n) = number of terms in row n of A214850.

2, 3, 6, 3, 3, 5, 3, 5, 4, 6, 4, 5, 4, 3, 8, 2, 3, 6, 3, 8, 6, 3, 8, 6, 2, 6, 8, 3, 4, 4, 4, 6, 4, 3, 8, 8, 2, 3, 7, 3, 8, 12, 3, 4, 8, 2, 8, 8, 3, 4, 8, 4, 8, 9, 4, 10, 6, 2, 5, 8, 2, 8, 6, 4, 6, 5, 3, 8, 4, 3, 8, 9, 4, 6, 7, 3, 8, 4, 4, 8, 5, 4, 8, 6, 5, 6

Offset: 1

Michel Lagneau, Apr 08 2013

nonn

Number of multiplicative finite groups G(p) with elements {T(2n+1,k)/pZ} where T(2n+1,k) is the reduced trajectory of the Collatz problem whose elements are all odd and p <= A075684(n) + 1.

			a(18) = 6 because there exist 6 finite groups given by row 18 of A214850 where p = 2, 4, 6, 8, 12 and 18. The Collatz trajectory of the number 2*18 + 1 = 37 with odd numbers is T(37,k) = {37, 7, 11, 17, 13, 5, 1}, and the 6 groups G(p) are:
G(2) = {T(37,k)/2Z} = {1}
G(4) = {T(37,k)/4Z} = {1, 3}
G(6) = {T(37,k)/6Z} = {1, 5}
G(8) = {T(37,k)/8Z}  = {1, 3, 5, 7}
G(12) = {T(37,k)/12Z} = {1, 5, 7, 11}
G(18) = {T(37,k)/18Z} = {1, 5, 7, 11, 13, 17}
G(18) is a cyclic group because the element 5 (or 11) generates the group:
5^1 == 5, 5^2 == 7, 5^3 == 17, 5^4 == 13, 5^5 == 11, 5^6 == 1 (mod 18).
G(8) is not a cyclic group.
a(170) = 32 because there exist 32 finite groups with two elements given by row 170 of A214850 where p = 2, 4, 6, 8, 10, 12, 18, 20, 24, 30, 34, 36, 38, 40, 60, 68, 72, 76, 90, 102, 114, 120, 136, 152, 170, 180, 190, 204, 228, 306, 340, 342. The Collatz trajectory of the number 2*170 + 1 = 341 with odd numbers is T(341,k) = {1, 341}.

Michel Lagneau, Table of n, a(n) for n = 1..500

Cf. A075680, A075684, A214850.

A267435 Numbers n such that each reduced Collatz trajectory (mod p): (n, T(n), T(T(n)),..., 4, 2, 1) / pZ, where the odd prime p is the number of iterations needed to reach 1, contains exactly the p-1 values {1, 2, 3, .., p-1}.

Original entry on oeis.org

8, 20, 32, 320, 2048, 2216, 8192, 13312, 87040, 218432, 524288, 89478400, 536870912, 137438953472, 250199979283796, 9007199254740992, 63800994005254144, 96076791692656640, 382805968326492160, 576460752303423488, 2305843009213693952, 4099276399740365440

Offset: 1

Michel Lagneau, Jan 15 2016

nonn

Or numbers n such that the multiplicative groups {n, T(n), T(T(n)),..., 4, 2, 1} / pZ are of order p-1.
Property of the sequence:
This sequence provides a link with Artin’s conjecture on primitive roots.
Conjecture: the sequence is infinite (corollary of a Artin’s conjecture  because the sequence contains the numbers 2^A001122(k) where A001122 are the primes with primitive root 2).
The sequence is divided into two class of numbers:
i) A class of powers of 2: 2^3, 2^5, 2^11, 2^13, 2^19, 2^29, 2^37, 2^53, ..., 2^A001122(k),…
ii) A class of non-powers of 2: 20, 320, 2216, 13312, 87040, 218432, 89478400...
The corresponding p of the sequence are 3, 7, 5, 11, 11, 19, 13, 19, 19, 23, 19, 29,...

			20 is in the sequence because the Collatz trajectory of 20 is {20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1} with 7 iterations, and the corresponding reduced trajectory (mod 7) is {6, 3, 5, 2, 1, 4, 2, 1} => the multiplicative group of order 6 is G = {1, 2, 3, 4, 5, 6}.

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..37
Wikipedia, Artin's conjecture on primitive roots.

Cf. A001122, A006667, A214850.

nn:=10000:T:=array(1..2000):U:=array(1..2000):
for n from 1 to 10000000 do:
  kk:=1:m:=n:T[kk]:=n:it:=0:
    for i from 1 to nn while(m<>1) do:
     if irem(m,2)=0
       then
       m:=m/2:kk:=kk+1:T[kk]:=m:it:=it+1:
       else
       m:=3*m+1:kk:=kk+1:T[kk]:=m:it:=it+1:
     fi:
    od:
      if isprime(it)
       then
       lst:={}:
       for p from 1 to it do:
        lst:=lst union {irem(T[p],it)}:
       od:
        n0:=nops(lst):
        if n0=it-1 and lst[1]=1
         then
         print(n):
         else
        fi:
      fi:
    od:

a(14)-a(22) from Hiroaki Yamanouchi, Jan 19 2016

cp's OEIS Frontend

A224504 a(n) = number of terms in row n of A214850.

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