A196871 -id:A196871 - cp's OEIS frontend

A078350 Number of primes in {n, f(n), f(f(n)), ..., 1}, where f is the Collatz function defined by f(x) = x/2 if x is even; f(x) = 3x + 1 if x is odd.

0, 1, 3, 1, 2, 3, 6, 1, 6, 2, 5, 3, 3, 6, 4, 1, 4, 6, 7, 2, 1, 5, 4, 3, 7, 3, 25, 6, 6, 4, 24, 1, 7, 4, 3, 6, 7, 7, 11, 2, 25, 1, 8, 5, 4, 4, 23, 3, 7, 7, 6, 3, 3, 25, 24, 6, 8, 6, 11, 4, 5, 24, 20, 1, 7, 7, 9, 4, 3, 3, 22, 6, 25, 7, 2, 7, 6, 11, 11, 2, 5, 25, 24, 1, 1, 8, 9, 5, 10, 4, 20, 4, 3, 23, 20

Offset: 1

Joseph L. Pe, Dec 23 2002

Number of primes in the trajectory of n under the 3x+1 map (i.e., the number of primes until the trajectory reaches 1, including 2 once). - Benoit Cloitre, Dec 23 2002
a(A196871(n)) = 0. - Reinhard Zumkeller, Oct 08 2011
a(A181921(n)) = n and a(m) <> n for m < A181921(n). - Reinhard Zumkeller, Apr 03 2012

			3 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1; in this trajectory 3, 5, 2 are primes hence a(3) = 3. - _Benoit Cloitre_, Dec 23 2002
The finite sequence n, f(n), f(f(n)), ..., 1 for n = 12 is 12, 6, 3, 10, 5, 16, 8, 4, 2, 1, which has three prime terms. Hence a(12) = 3.

T. D. Noe, Table of n, a(n) for n = 1..10000
J. C. Lagarias, The 3x+1 problem and its generalizations, Amer. Math. Monthly, 92 (1985), 3-23.
Eric Weisstein's World of Mathematics, Collatz Problem
Wikipedia, Collatz conjecture
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A064684, A006370, A010051.

a078350 n = sum $ map a010051 $ takeWhile (> 1) $ iterate a006370 n  -- Reinhard Zumkeller, Oct 08 2011

a:= proc(n) option remember; `if`(n=1, 0,
     `if`(isprime(n), 1, 0)+a(`if`(n::even, n/2, 3*n+1)))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 04 2024

f[n_] := n/2 /; Mod[n, 2] == 0 f[n_] := 3 n + 1 /; Mod[n, 2] == 1 g[n_] := Module[{i, p}, i = n; p = 0; While[i > 1, If[PrimeQ[i], p = p + 1]; i = f[i]]; p]; Table[g[n], {n, 1, 100}]
Table[Count[NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#!=1&],?PrimeQ],{n,100}] (* _Harvey P. Dale, Aug 29 2012 *)

for(n=2,500,s=n; t=0; while(s!=1,if(isprime(s)==1,t=t+1,t=t); if(s%2==0,s=s/2,s=(3*s+1)); if(s==1,print1(t,","); ); )) \\ Benoit Cloitre, Dec 23 2002

a(n)=my(s=isprime(n));while(n>1,if(n%2,n=(3*n+1)/2,n/=2);s+=isprime(n));s \\ Charles R Greathouse IV, Apr 28 2015

A078350(n,c=n>1)={while(1>=valuation(n,2), isprime(n)&&c++; n=n*3+1);c} \\ M. F. Hasler, Dec 05 2017

a(n) = A055509(n) + 1 for n > 1.

a(n) = 1 when n > 1 is in A000079, i.e., a power of 2. - Benoit Cloitre, Dec 20 2017

Edited by N. J. A. Sloane, Jan 17 2009 at the suggestion of R. J. Mathar

A055509 Number of odd primes in sequence obtained in 3x+1 (or Collatz) problem starting at n.

Original entry on oeis.org

0, 0, 2, 0, 1, 2, 5, 0, 5, 1, 4, 2, 2, 5, 3, 0, 3, 5, 6, 1, 0, 4, 3, 2, 6, 2, 24, 5, 5, 3, 23, 0, 6, 3, 2, 5, 6, 6, 10, 1, 24, 0, 7, 4, 3, 3, 22, 2, 6, 6, 5, 2, 2, 24, 23, 5, 7, 5, 10, 3, 4, 23, 19, 0, 6, 6, 8, 3, 2, 2, 21, 5, 24, 6, 1, 6, 5, 10, 10, 1, 4, 24, 23, 0, 0, 7, 8, 4, 9, 3, 19, 3, 2, 22, 19

Offset: 1

G. L. Honaker, Jr., Jun 30 2000

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Collatz Problem
Wikipedia, Collatz conjecture
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A055510.

Cf. A006370, A010051.

a055509 n = sum $ map a010051 $ takeWhile (> 2) $ iterate a006370 n -- Reinhard Zumkeller, Oct 08 2011

g:= proc(n) option remember;
   local x;
   x:= 3*n+1;
   x:= x/2^padic:-ordp(x,2);
   if isprime(n) then procname(x)+1 else procname(x) fi
end proc:
g(1):= 0:
seq(g(n/2^padic:-ordp(n,2)),n=1..100); # Robert Israel, Dec 05 2017

Join[{0}, Table[Count[NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &], ?PrimeQ] - 1, {n, 2, 94}]] (* _Jayanta Basu, Jun 15 2013 *)

A078350(n,c=0)={while(1>=valuation(n,2), isprime(n)&&c++; n=n*3+1);c} \\ M. F. Hasler, Dec 05 2017

a(n) = A078350(n) - 1 for n > 1.
a(A196871(n)) = 0. - Reinhard Zumkeller, Oct 08 2011
From Robert Israel, Dec 05 2017: (Start)
If n is odd, a(n) = a(3*n+1) + A010051(n).
If n is even, a(n) = a(n/2). (End)

More terms from Larry Reeves (larryr(AT)acm.org), Aug 09 2001

A221475 Odd numbers having no odd primes in their Collatz (3x+1) trajectory.

Original entry on oeis.org

1, 21, 85, 341, 453, 909, 1365, 1813, 5461, 7281, 9669, 21845, 29013, 29125, 38835, 45839, 54327, 58197, 58253, 68759, 77667, 81491, 87381, 103139, 116053, 116501, 122237, 154709, 154737, 155341, 183357, 232789, 233013, 257551, 275037, 310385, 310669, 325965

Offset: 1

T. D. Noe, Feb 14 2013

nonn

Sequence A196871 contains these terms and even numbers that are 2^k times terms in this sequence.

Cf. A196871.

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; t = {1}; n = 1; While[Length[t] < 50, n = n + 2; If[Union[Drop[PrimeQ[Collatz[n]], -2]] == {False}, AppendTo[t, n]]]; t

A078440 Numbers n with property that n is not a power of 2 and the finite sequence n, f(n), f(f(n)), ...., 1 in the Collatz (or 3x + 1) problem contains exactly one prime. (The earliest "1" is meant.)

Original entry on oeis.org

21, 42, 84, 85, 168, 170, 336, 340, 341, 453, 672, 680, 682, 906, 909, 1344, 1360, 1364, 1365, 1812, 1813, 1818, 2688, 2720, 2728, 2730, 3624, 3626, 3636, 5376, 5440, 5456, 5460, 5461, 7248, 7252, 7272, 7281, 9669

Offset: 1

Joseph L. Pe, Dec 31 2002

nonn

f(n) = n/2 if n is even, = 3n + 1 if n is odd. Powers 2^n trivially have exactly one prime in n, f(n), f(f(n)), ..., 2, 1, namely 2 and so are excluded from the sequence.

A055509(a(n)) = 0; A078350(a(n)) <= 1.

			n, f(n), f(f(n)), .... for n = 21 is: 21, 64, 32, 16, 8, 4, 2, 1, which has exactly one prime, that is, 2. Hence 21 belongs to the sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Collatz Problem
Wikipedia, Collatz conjecture
Index entries for sequences related to 3x+1 (or Collatz) problem

A006370; subsequence of A196871 (with binary powers).

a078440 n = a078440_list !! (n-1)
a078440_list = filter notbp a196871_list where
   notbp x = m > 0 && x > 1 || m == 0 && notbp x' where
      (x',m) = divMod x 2
-- Reinhard Zumkeller, Oct 08 2011

f[n_] := n/2 /; Mod[n, 2] == 0 f[n_] := 3 n + 1 /; Mod[n, 2] == 1 g[n_] := Module[{i, p}, i = n; p = 0; While[i > 1, If[PrimeQ[i], p = p + 1]; i = f[i]]; p]; Select[Range[10^4], g[ # ] == 1 && ! IntegerQ[Log[2, # ]] &]
pQ[n_]:=Count[NestWhileList[If[EvenQ[#],#/2,3#+1]&,n, #>1&], ?PrimeQ] == 1; With[ {nn=10000},Complement[Select[Range[nn],pQ],2^Range[Floor[ Log[ 2,nn]]]]] (* _Harvey P. Dale, Oct 19 2011 *)

A362959 Numbers k such that the Collatz orbit that begins with k does not contain an odd prime afterwards.

Original entry on oeis.org

4, 5, 8, 16, 21, 32, 42, 64, 84, 85, 113, 128, 168, 170, 227, 256, 336, 340, 341, 453, 512, 672, 680, 682, 906, 909, 1024, 1344, 1360, 1364, 1365, 1812, 1813, 1818, 2048, 2417, 2688, 2720, 2728, 2730, 3624, 3626, 3636, 3637, 4096, 5376, 5440, 5456, 5460, 5461, 7248

Offset: 1

Hugo Pfoertner, May 18 2023

nonn

			a(11) = 113, because its Collatz orbit 340, 170, 85, 256, 128, 64, 32, 16, 8, 4, 2 avoids odd primes.

Hugo Pfoertner, Table of n, a(n) for n = 1..10010

Cf. A000079, A006370, A006577, A196871.

Select[Range[4, 7500], NoneTrue[Rest@ NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, #, # > 1 &], And[OddQ[#], PrimeQ[#]] &] &] (* Michael De Vlieger, May 18 2023 *)

cm(k) = {while (k>1, k=if(k%2,3*k+1,k/2); if(isprime(k),break)); k};
for (k=3, 7500, if(cm(k)==2, print1(k,", ")))

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:

cp's OEIS Frontend

A078350 Number of primes in {n, f(n), f(f(n)), ..., 1}, where f is the Collatz function defined by f(x) = x/2 if x is even; f(x) = 3x + 1 if x is odd.

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A055509 Number of odd primes in sequence obtained in 3x+1 (or Collatz) problem starting at n.

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A221475 Odd numbers having no odd primes in their Collatz (3x+1) trajectory.

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A078440 Numbers n with property that n is not a power of 2 and the finite sequence n, f(n), f(f(n)), ...., 1 in the Collatz (or 3x + 1) problem contains exactly one prime. (The earliest "1" is meant.)

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A362959 Numbers k such that the Collatz orbit that begins with k does not contain an odd prime afterwards.

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Mathematica

PARI

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

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Maple