A078350 -id:A078350 - cp's OEIS frontend

A070983 Erroneous version of A078350.

1, 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

Offset: 1

dead

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

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

A196871 Numbers having no odd primes in their Collatz (3x+1) trajectory.

Original entry on oeis.org

1, 2, 4, 8, 16, 21, 32, 42, 64, 84, 85, 128, 168, 170, 256, 336, 340, 341, 453, 512, 672, 680, 682, 906, 909, 1024, 1344, 1360, 1364, 1365, 1812, 1813, 1818, 2048, 2688, 2720, 2728, 2730, 3624, 3626, 3636, 4096, 5376, 5440, 5456, 5460, 5461, 7248, 7252, 7272

Offset: 1

Reinhard Zumkeller, Oct 08 2011

nonn

Union of A000079 and A078440;

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

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

Cf. A006370; subsequence of A065090.

Cf. A221475 (odd numbers in this sequence).

a196871 n = a196871_list !! (n-1)
a196871_list = filter
   (all (== 0) . map a010051 . takeWhile (> 2) . iterate a006370) [1..]

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; Join[{1, 2}, Select[Range[3, 10000], Union[Drop[PrimeQ[Collatz[#]], -2]] == {False} &]] (* T. D. Noe, Jan 22 2013 *)

A078373 n sets a record for the 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.

Original entry on oeis.org

2, 3, 7, 19, 27, 97, 171, 231, 487, 763, 1071, 4011, 6171, 10971, 17647, 47059, 99151, 117511, 202471, 260847, 481959, 963919, 1564063, 1805311, 1993215, 6991599, 8400511, 11200681, 36791535, 46564287, 103359483, 206718967, 359502063

Offset: 1

Joseph L. Pe, Dec 24 2002

nonn

			The sequence n, f(n), f(f(n)), ..., 1 for n = 7 is: 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, which has six prime terms, more prime terms than for any n < 7. Hence 7 sets a record and so belongs to the sequence.

Jud McCranie, Table of n, a(n) for n = 1..40
J. C. Lagarias, The 3x+1 problem and its generalizations, Amer. Math. Monthly, 92 (1985), 3-23.
Carlos Rivera, Puzzle 634. Primes in Collatz trajectory
Carlos Rivera, Puzzle 922. Follow up to Puzzle 634

A362958 gives the corresponding numbers of primes.

Cf. A055509, A181921.

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]; high = 0; a = {}; For[j = 1, j <= 10^5, j++, k = g[j]; If[k > high, high = k; a = Append[a, j]]]; a
(* Second program: *)
With[{s = Array[Count[NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, #, # > 1 &], ?PrimeQ] &, 10^5]}, Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]] (* _Michael De Vlieger, Apr 21 2018 *)

A078350(n)=my(s=isprime(n)); while(n>1, if(n%2, n=(3*n+1)/2, n/=2); s+=isprime(n)); s
r=0; for(n=2,1e9, t=A078350(n); if(t>r, r=t; print1(n", "))) \\ Charles R Greathouse IV, Apr 28 2015

a(18)-a(30) from Donovan Johnson, Jul 02 2010

a(31)-a(33) from Carlos Rivera, Apr 15 2012

A177000 The Collatz iteration of these primes produces only even numbers, primes and 1.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 29, 37, 53, 59, 67, 89, 101, 131, 149, 157, 179, 181, 197, 241, 269, 277, 349, 397, 739, 853, 1109, 1237, 1429, 1621, 1861, 1877, 2161, 2389, 2531, 2957, 3413, 3797, 4549, 5717, 7621, 10069, 13397, 17749, 20021, 31541, 40277

Offset: 1

T. D. Noe, Apr 30 2010

The Collatz iteration of primes of the form (10*4^k-1)/3 produces only one additional prime: 5. The Collatz iteration of primes of the form (13*4^k-1)/3 produces only two additional primes: 5 and 13. This sequence is probably infinite.

In a sense, these are the simplest Collatz iterations starting with a prime number. Except for the increases (3x+1) when an odd prime occurs, the sequence produced by starting with a(n) is decreasing. All the primes that occur in such a Collatz iteration are in this sequence. - T. D. Noe, Oct 05 2011

			The Collatz iteration of 7 produces 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, and 1, which are either even, prime, or 1.

Donovan Johnson and T. D. Noe, Table of n, a(n) for n = 1..10000 (Donovan Johnson to 203 terms)
Eric Weisstein's World of Mathematics, Collatz Problem
Wikipedia, Collatz conjecture
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A055509, A078350, A078373, A070165, A010051.

a177000 n = a177000_list !! (n-1)
a177000_list = filter (all (\x -> even x || a010051' x == 1) .
                       (init . a070165_row)) a000040_list
-- Reinhard Zumkeller, Apr 03 2012

Collatz[n_] := NestWhileList[If[EvenQ[ # ], #/2, 3#+1] &, n, #>1 &]; Reap[Do[p=Prime[n]; s=Most[Select[Collatz[p],OddQ]]; If[And@@PrimeQ[s], Sow[p]], {n,PrimePi[10^4]}]][[2,1]]
oenQ[n_]:=AllTrue[DeleteCases[Most[NestWhileList[If[EvenQ[#],#/2, 3#+1]&,n, #>1&]], ?PrimeQ],EvenQ]; Select[Prime[Range[5000]], oenQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale, Dec 28 2014 *)

is(n)=isprime(n) && (n<23 || is((3*n+1)>>valuation(3*n+1,2))) \\ Charles R Greathouse IV, Jun 20 2013

A181921 The smallest positive integer that produces exactly n primes in a Collatz trajectory.

Original entry on oeis.org

2, 5, 3, 15, 11, 7, 19, 43, 67, 89, 39, 127, 123, 223, 111, 351, 175, 155, 103, 63, 107, 71, 47, 31, 27, 97, 193, 171, 231, 487, 1087, 763, 2223, 2143, 1263, 1071, 4011, 6919, 8127, 13183, 6591, 6943, 6171, 10971, 46443, 48927, 35295, 17647, 70589, 47059

Offset: 1

G. L. Honaker, Jr., Apr 01 2012

nonn

			a(6) = 7 because the Collatz trajectory of 7 is {7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1}, containing 6 primes {7, 11, 17, 13, 5, 2}, and 7 is the smallest positive integer for which exactly 6 primes occur via this trajectory.

Reinhard Zumkeller and Jud McCranie, Table of n, a(n) for n = 1..92 (first 75 numbers from Reinhard Zumkeller)
Eric Weisstein's World of Mathematics, Collatz Problem
Wikipedia, Collatz conjecture
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A055509, A078350.

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a181921 = (+ 1) . fromJust . (`elemIndex` a078350_list)
-- Reinhard Zumkeller, Apr 03 2012

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; nn = 50; t = Table[0, {nn}]; t[[1]] = 2; todo = nn - 1; n = 3; While[todo > 0, ps = Length[Select[Collatz[n], PrimeQ]]; If[ps <= nn && t[[ps]] == 0, t[[ps]] = n; todo--]; n = n + 2]; t (* T. D. Noe, Apr 02 2012 *)
With[{lst=Table[Count[NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&],?PrimeQ],{n, 71000}]},Table[Position[lst,k,1,1],{k,50}]//Flatten] (* _Harvey P. Dale, Sep 08 2018 *)

np(n)=my(t=1);while(n>2,t+=isprime(n);if(n%2,n+=n>>1+1,n>>=1));t
v=vector(40);n=1;while(1,t=np(n++);if(t<=#v&&v[t]==0, v[t]=n; if(vecmin(v), return(v)))) \\ Charles R Greathouse IV, Apr 01 2012

a(13)-a(50) from Charles R Greathouse IV, Apr 01 2012

A275866 Number of semiprimes in {n, f(n), f(f(n)), ...., 1}, where f is the Collatz function.

Original entry on oeis.org

0, 0, 2, 1, 1, 3, 5, 1, 7, 2, 4, 3, 2, 6, 6, 1, 3, 7, 6, 2, 2, 5, 4, 3, 8, 3, 38, 6, 5, 6, 36, 1, 9, 4, 4, 7, 6, 7, 12, 2, 37, 2, 9, 5, 4, 5, 35, 3, 8, 8, 8, 3, 2, 38, 38, 6, 11, 6, 10, 6, 5, 37, 36, 1, 9, 9, 8, 4, 4, 4, 34, 7, 38, 7, 3, 7, 7, 12, 11, 2, 6, 38

Offset: 1

Michel Lagneau, Aug 11 2016

nonn

Number of semiprimes in the trajectory of n under the 3x+1 map (i.e. the number of semiprimes until the trajectory reaches 1).

It seems that about 15% of the terms satisfy a(i) = a(i+1). For example, up to 100000, 15140 terms satisfy this condition.

			a(9)=7 because the trajectory of 9 is 9 -> 28 -> 14 -> 7 -> 22 -> 11 -> 34 -> 17 -> 52 -> 26 -> 13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1 and the 7 semiprimes of this trajectory are 9, 14, 22, 34, 26, 10 and 4.

Michel Lagneau, Table of n, a(n) for n = 1..10000

Cf. A006577, A078350.

Table[Count[NestWhileList[If[EvenQ@ #, #/2, 3 # + 1] &, n, # != 1 &], k_ /; PrimeOmega@ k == 2], {n, 82}] (* Michael De Vlieger, Aug 11 2016 *)

print1(0, ", ");for(n=2, 100, s=n; t=0; while(s!=1, if(bigomega(s)==2 , t=t+1, t=t); if(s%2==0, s=s/2, s=(3*s+1)); if(s==1, print1(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 *)

A068684 Primes obtained as a concatenation p,q,p where p and q are successive primes and p

Original entry on oeis.org

353, 131713, 171917, 192319, 293129, 374137, 434743, 596159, 677167, 139149139, 163167163, 179181179, 223227223, 229233229, 269271269, 281283281, 347349347, 379383379, 547557547, 683691683, 761769761, 857859857, 863877863, 102110311021, 103910491039, 108710911087, 109110931091, 109310971093

Offset: 1

Amarnath Murthy, Mar 02 2002

			171917 is a prime which is the concatenation of 17, 19 and 17.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A068683, A078350.

cat3:= proc(a,b,c) local alpha,beta;
    beta:= ilog10(c)+1;
    alpha:= beta + ilog10(b)+1;
    10^alpha*a + 10^beta*b + c
end proc:
R:= NULL: count:= 0: q:= 2:
while count < 100 do
  p:= q; q:= nextprime(q);
  v:= cat3(p,q,p);
  if isprime(v) then R:= R,v; count:= count+1;
  fi
od:
R; # Robert Israel, Jul 01 2025

f(n)=prime(n)*(10^(ceil(log(prime(n+1))/log(10))+ceil(log(prime(n))/log(10))))+ prime(n+1)*10^ceil(log(prime(n))/log(10))+prime(n);
for(n=1,300, if(isprime(f(n))==1, print1(f(n),", ")))

More terms from Benoit Cloitre, Mar 21 2002

More terms from Robert Israel, Jul 02 2025

A267830 Number of nonprime numbers 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.

Original entry on oeis.org

1, 1, 5, 2, 4, 6, 11, 3, 14, 5, 10, 7, 7, 12, 14, 4, 9, 15, 14, 6, 7, 11, 12, 8, 17, 8, 87, 13, 13, 15, 83, 5, 20, 10, 11, 16, 15, 15, 24, 7, 85, 8, 22, 12, 13, 13, 82, 9, 18, 18, 19, 9, 9, 88, 89, 14, 25, 14, 22, 16, 15, 84, 88, 6, 21, 21, 19, 11, 12, 12, 81

Offset: 1

Michel Lagneau, Jan 21 2016

nonn

Number of nonprime numbers in the trajectory of n under the 3x+1 map (i.e., the number of nonprime numbers until the trajectory reaches 1).

It seems that about 20% of the terms satisfy a(i) = a(i+1). For example, up to 10^6, 201085 terms satisfy this condition.

			a(9)=14 because the trajectory of 9 is 9 -> 28 -> 14 -> 7 -> 22 -> 11 -> 34 -> 17 -> 52 -> 26 -> 13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1 and the 14 nonprimes of this trajectory are 9, 28, 14, 22, 34, 52, 26, 40, 20, 10, 16, 8, 4, and 1.

Michel Lagneau, Table of n, a(n) for n = 1..10000

Cf. A006577, A008908, A078350, A018252.

A267830[n_] := Count[NestWhileList[If[EvenQ@#, #/2, 3 # + 1] &, n, # != 1 &], ?(Not@PrimeQ@# &)] (* _JungHwan Min, Jan 24 2016 *)

for(n=1, 100, s=n; t=0; while(s!=1, if(!isprime(s) , t++); if(s%2==0, s=s/2, s=(3*s+1)); if(s==1, print1(t+1, ", "); ); ))

a(n)= A008908(n) - A078350(n).

cp's OEIS Frontend

A070983 Erroneous version of A078350.

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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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A196871 Numbers having no odd primes in their Collatz (3x+1) trajectory.

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A078373 n sets a record for the 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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A177000 The Collatz iteration of these primes produces only even numbers, primes and 1.

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Haskell

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A181921 The smallest positive integer that produces exactly n primes in a Collatz trajectory.

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A275866 Number of semiprimes in {n, f(n), f(f(n)), ...., 1}, where f is the Collatz function.

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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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