A133419 -id:A133419 - cp's OEIS frontend

A133420 Number of steps to reach 1 under repeated applications of the "5x+1" map of A133419, or -1 if 1 is never reached.

0, 1, 1, 2, 14, 2, 5, 3, 2, 15, 9, 3, 12, 6, 15, 4, 9, 3, 7, 16, 6, 10, 21, 4, 9, 13, 3, 7, 18, 16, 16, 5, 10, 10, 14, 4, 19, 8, 13, 17, 13, 7, 7, 11, 16, 22, 26, 5, 16, 10, 10, 14, 25, 4, 25, 8, 8, 19, 23, 17, 13, 17, 7, 6, 17, 11, 11, 11, 22, 15, 26, 5, 16, 20, 10, 9, 20, 14, 14, 18, 4, 14

Offset: 1

N. J. A. Sloane, Nov 27 2007

The 5x+1 map sends x to x/2 if x is even, x/3 if x is divisible by 3, otherwise 5x+1.

R. J. Mathar, Table of n, a(n) for n = 1..20000
R. J. Mathar, Maple program to make b-file
Tomás Oliveira e Silva, The px+1 problem
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A133419.

A133423 Analog of A006684 for the 5x+1 problem (cf. A133419).

Original entry on oeis.org

1, 5, 17, 23, 41, 53, 65, 71, 77, 95, 221, 317, 365, 383, 3317, 3575, 3605, 6473, 24125, 31901, 39965, 44183, 163733, 317885, 490541, 519113, 558365, 602591, 707735, 753023, 1019615, 1463897, 1597973, 1752575, 4595735, 6197855

Offset: 1

N. J. A. Sloane, Nov 27 2007

nonn

The 5x+1 map sends x to x/2 if x is even, x/3 if x is divisible by 3, otherwise 5x+1.

N. J. A. Sloane, Table of n, a(n) for n=1..89 (from the web page of Tomás Oliveira e Silva)
Tomás Oliveira e Silva, The px+1 problem
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A133419, A133424, A133419, ...

A133424 Analog of A060410 for the 5x+1 problem (cf. A133419).

Original entry on oeis.org

6, 66, 216, 366, 516, 666, 816, 1116, 2016, 4866, 5766, 8616, 9516, 229866, 286116, 473616, 737016, 3230766, 4438266, 6260016, 10637016, 107662496, 117661116, 152291166, 176254866, 179900766, 201566166, 230949516

Offset: 1

N. J. A. Sloane, Nov 27 2007

nonn

The 5x+1 map sends x to x/2 if x is even, x/3 if x is divisible by 3, otherwise 5x+1.

N. J. A. Sloane, Table of n, a(n) for n=1..89 (from the web page of Tomás Oliveira e Silva)
Tomás Oliveira e Silva, The px+1 problem
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A133419, A133423, A133419, ...

A232711 Conjectured list of numbers whose trajectory under the '5x+1' map eventually reaches 1.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 15, 16, 19, 24, 30, 32, 38, 48, 51, 60, 64, 65, 76, 96, 97, 102, 120, 128, 130, 137, 152, 155, 163, 175, 192, 194, 204, 219, 240, 243, 256, 260, 274, 304, 307, 310, 326, 343, 350, 384, 388, 397, 408, 417, 429, 438, 480, 486, 491, 512

Offset: 1

Jon Perry, Nov 28 2013

nonn

This is conjectural in that there is no known proof that 7, 9, 11, etc. (see A267970) do not eventually cycle. - N. J. A. Sloane, Jan 23 2016
It appears that most numbers diverge, but nothing is known for certain.
Note that the computer programs do not actually calculate a complete list of "numbers k such that the Collatz-like map T: if x odd, x -> 5*x+1 and if x even, x -> x/2, when started at k, eventually reaches 1".

			Beginning with 15 we get the trajectory 15, 76, 38, 19, 96, 48, 24, 12, 6, 3, 16, 8, 4, 2, 1, so 15 is a term.

Dmitry Kamenetsky, Table of n, a(n) for n = 1..1000
Tomás Oliveira e Silva, The px+1 problem.
Index entries for sequences related to 3x+1 (or Collatz) problem

See A267969, A267970 for other trajectories under this map T.
Cf. A057688, A133419.
Cf. A070165 (usual Collatz iteration).

for (i=1;i<2000;i++) {
c=0;
n=i;
while (n>1) {c++;if (n%2==0) n/=2; else n=5*n+1;if (c>100) break;}
if (c<101) document.write(i+", ");
} (Warning: bad program - will not find all the terms. - N. J. A. Sloane, Jan 23 2016)

cli=Compile[{{n,Integer}},If[OddQ[n],5n+1,n/2]//Round,RuntimeAttributes->{Listable},Parallelization->True]; okQ[n]:=Length[NestWhileList[cli,n, #>1&, 1,200]]<200; Select[Range[550],okQ] (* Harvey P. Dale, May 28 2014 *) (Warning: bad program - will not find all the terms. - N. J. A. Sloane, Jan 23 2016)

Entry revised (corrected definition, added warnings to programs, deleted b-file) by N. J. A. Sloane, Jan 23 2016

A133421 Image of n under one application of the "7x+1" map.

Original entry on oeis.org

8, 1, 1, 2, 1, 3, 50, 4, 3, 5, 78, 6, 92, 7, 5, 8, 120, 9, 134, 10, 7, 11, 162, 12, 5, 13, 9, 14, 204, 15, 218, 16, 11, 17, 7, 18, 260, 19, 13, 20, 288, 21, 302, 22, 15, 23, 330, 24, 344, 25, 17, 26, 372, 27, 11, 28, 19, 29, 414, 30, 428, 31, 21, 32, 13, 33, 470, 34, 23, 35, 498

Offset: 1

N. J. A. Sloane, Nov 27 2007

The 7x+1 map sends x to x/2 if x is even, x/3 if x is odd and divisible by 3, x/5 if x is not divisible by 6 and divisible by 5, otherwise 7x+1.

Harvey P. Dale, Table of n, a(n) for n = 1..10000
Tomás Oliveira e Silva, The px+1 problem
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A133422, A133419.

Table[Nest[Which[EvenQ[#],#/2,Divisible[#,3],#/3,Divisible[#,5],#/5, True, 7#+1]&,n,1],{n,75}] (* Harvey P. Dale, Nov 05 2011 *)

a(n)=if(n%2,if(n%3,if(n%5,7*n+1,n/5),n/3),n/2) \\ Charles R Greathouse IV, Sep 02 2015

from _future_ import division
def A133421(n):
    return n//2 if not n % 2 else (n//3 if not n % 3 else (n//5 if not n % 5 else 7*n+1)) # Chai Wah Wu, Mar 04 2018

From Chai Wah Wu, Mar 04 2018: (Start)
a(n) = 2*a(n-30) - a(n-60) for n > 60.
G.f.: x*(6*x^58 + x^57 + x^56 + 2*x^55 + x^54 + 3*x^53 + 48*x^52 + 4*x^51 + 3*x^50 + 5*x^49 + 76*x^48 + 6*x^47 + 90*x^46 + 7*x^45 + 5*x^44 + 8*x^43 + 118*x^42 + 9*x^41 + 132*x^40 + 10*x^39 + 7*x^38 + 11*x^37 + 160*x^36 + 12*x^35 + 5*x^34 + 13*x^33 + 9*x^32 + 14*x^31 + 202*x^30 + 15*x^29 + 204*x^28 + 14*x^27 + 9*x^26 + 13*x^25 + 5*x^24 + 12*x^23 + 162*x^22 + 11*x^21 + 7*x^20 + 10*x^19 + 134*x^18 + 9*x^17 + 120*x^16 + 8*x^15 + 5*x^14 + 7*x^13 + 92*x^12 + 6*x^11 + 78*x^10 + 5*x^9 + 3*x^8 + 4*x^7 + 50*x^6 + 3*x^5 + x^4 + 2*x^3 + x^2 + x + 8)/(x^60 - 2*x^30 + 1). (End)

More terms from Sean A. Irvine, Mar 29 2010

Comment clarified by Chai Wah Wu, Mar 04 2018

A133425 Analog of A006684 for the 7x+1 problem (cf. A133421).

Original entry on oeis.org

1, 7, 11, 19, 31, 43, 163, 283, 403, 1111, 1123, 1243, 1303, 1549, 1963, 4123, 9643, 10003, 11539, 21431, 76963, 97031, 468109, 1351963, 4553323, 4778471, 5163139, 6563551, 7618843, 45214123, 65704243, 161738803, 202903723

Offset: 1

N. J. A. Sloane, Nov 27 2007

nonn

The 7x+1 map sends x to x/2 if x is even, x/3 if x is divisible by 3, x/5 if x is divisible by 5, otherwise 7x+1.

N. J. A. Sloane, Table of n, a(n) for n=1..53 (from the web page of Tomás Oliveira e Silva)
Tomás Oliveira e Silva, The px+1 problem
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A133421, A133426, A133419, ...

A133426 Analog of A060410 for the 7x+1 problem (cf. A133421).

Original entry on oeis.org

8, 50, 162, 470, 9584, 28400, 91890, 193040, 265070, 291824, 337100, 388830, 524070, 3231824, 18052070, 30949850, 31021880, 65552768, 774659600, 10276607888, 38128783428, 190067750300, 4835458627140, 25515567812750

Offset: 1

N. J. A. Sloane, Nov 27 2007

nonn

The 7x+1 map sends x to x/2 if x is even, x/3 if x is divisible by 3, x/5 if x is divisible by 5, otherwise 7x+1.

N. J. A. Sloane, Table of n, a(n) for n=1..53 (from the web page of Tomás Oliveira e Silva)
Tomás Oliveira e Silva, The px+1 problem
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A133421, A133425, A133419, ...

A133422 Number of steps to reach 1 under repeated applications of the "7x+1" map of A133421, or -1 if 1 is never reached.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 4, 3, 2, 2, 12, 3, 9, 5, 2, 4, 6, 3, 21, 3, 5, 13, 6, 4, 2, 10, 3, 6, 10, 3, 47, 5, 13, 7, 5, 4, 13, 22, 10, 4, 8, 6, 46, 14, 3, 7, 16, 5, 50, 3, 7, 11, 51, 4, 13, 7, 22, 11, 10, 4, 9, 48, 6, 6, 10, 14, 19, 8, 7, 6, 17, 5, 10, 14, 3, 23, 7, 11, 17, 5, 4, 9, 14, 7, 7, 47, 11, 15

Offset: 1

N. J. A. Sloane, Nov 27 2007

The 7x+1 map sends x to x/2 if x is even, x/3 if x is divisible by 3, x/5 if x is divisible by 5, otherwise 7x+1.

Tomás Oliveira e Silva, The px+1 problem
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A133421, A133419, ...

nxt[x_]:= Which[Mod[x,2]==0,x/2,Mod[x,3]==0,x/3,Mod[x,5]==0,x/5,True,7x+1]; Table[First[First[Position[NestList[nxt,i,100],1]]]-1,{i,88}] (* Harvey P. Dale, Dec 27 2007 *)

More terms from Harvey P. Dale, Dec 27 2007

More terms from Sean A. Irvine, Mar 29 2010

A332018 a(n) = A038502(A000265(n)) if n is even or n == 0 (mod 3), a(n) = A038502(A000265(5*n + 1)) otherwise.

Original entry on oeis.org

1, 1, 1, 1, 13, 1, 1, 1, 1, 5, 7, 1, 11, 7, 5, 1, 43, 1, 1, 5, 7, 11, 29, 1, 7, 13, 1, 7, 73, 5, 13, 1, 11, 17, 11, 1, 31, 19, 13, 5, 103, 7, 1, 11, 5, 23, 59, 1, 41, 25, 17, 13, 133, 1, 23, 7, 19, 29, 37, 5, 17, 31, 7, 1, 163, 11, 7, 17, 23, 35, 89, 1, 61, 37

Offset: 1

Davis Smith, Feb 04 2020

a(n) is the greatest divisor of n coprime to 6 if n is not coprime to 6, otherwise a(n) is the greatest divisor of 5*n + 1 coprime to 6.
This is the '5x+1' map with the successive dividing steps removed. The 'Px+1' map with those steps removed: If x is divisible by any prime < P, then divide out those primes; otherwise multiply x by P, add 1, and then divide out the primes < P.
There is a conjecture which states that for any value of n > 0 there is a k such that a^{k}(n) = 1 or a^{k}(n) enters one of a finite number of periodic cycles, where a^{0}(n) = n and a^{k + 1}(n) = a(a^{k}(n)).

J. Lesieutre, On a Generalization of the Collatz Conjecture, Research Science Institute, 2007.
T. Oliveira e Silva, Computational verification of the 5x+1 and 7x+1 conjectures.

Cf. A000265, A038502, A133419.

[Gcd(n,6) ne 1 select n/(Gcd(n, 2^n)*Gcd(n, 3^n)) else (5*n + 1)/(Gcd(5*n + 1, 2^(5*n + 1))*Gcd(5*n + 1, 3^(5*n + 1))):n in [1..75]]; // Marius A. Burtea, Feb 06 2020

A332018 := proc(n) option remember;
if n mod 2 = 0 or n mod 3 = 0 then n/(2^padic[ordp](n, 2)*3^padic[ordp](n, 3))
else (5*n+1)/(2^padic[ordp](5*n+1, 2)*3^padic[ordp](5*n+1, 3)) fi end:
seq(A332018(n), n=1..80);

b[n_]:=Denominator[2^n/n]; c[n_]:=Denominator[3^n/n]; Table[If[EvenQ[n]||(Mod[n, 3] == 0), c[b[n]], c[b[5*n + 1]]], {n, 1, 80}]

A332018(n)=my(val(x)=x/(2^valuation(x,2)*3^valuation(x,3))); val(if(n%2&&n%3,5*n+1,n))

a(n) = A038502(A000265(A133419(n))).

a(n) = n/(gcd(n, 2^n)*gcd(n, 3^n)) if n is not coprime to 6, a(n) = (5*n + 1)/(gcd(5*n + 1, 2^(5*n + 1))*gcd(5*n + 1, 3^(5*n + 1))) otherwise.

A270968 Reduced 5x+1 function R applied to the odd integers: a(n) = R(2n-1), where R(k) = (5k+1)/2^r, with r as large as possible.

Original entry on oeis.org

3, 1, 13, 9, 23, 7, 33, 19, 43, 3, 53, 29, 63, 17, 73, 39, 83, 11, 93, 49, 103, 27, 113, 59, 123, 1, 133, 69, 143, 37, 153, 79, 163, 21, 173, 89, 183, 47, 193, 99, 203, 13, 213, 109, 223, 57, 233, 119, 243, 31, 253, 129, 263, 67, 273, 139, 283, 9, 293, 149, 303

Offset: 1

Michel Lagneau, Mar 27 2016

The odd-indexed terms a(2i+1) = 10i+3 = A017305(i), i>=0;
a(4i+4) = 10i+9 = A017377(i), i>=0;
a(8i+6) = 10i+7 = A017353(i), i>=0;
a(16i+2) = 10i+1 = A017281(i), i>=0.
Note that a(n) = a(16n-6) = a(6n-2)/3. No multiple of 5 is in this sequence.
a(n) = R(2n-1) < 2n-1 for n = 2, 6, 10, ..., 2+4i,...

			a(4)=9 because (2*4-1) = 7  -> (5*7+1)/2^2 = 9.

Cf. A075677, A017281, A017305, A017353, A017377, A133423, A133424, A174795, A133419, A133420, A232711, A259207, A267969.

nextOddK[n_] := Module[{m=5n+1}, While[EvenQ[m], m=m/2]; m]; (* assumes odd n *) Table[nextOddK[n], {n, 1, 200, 2}]

a(n) = my(m = 2*n-1, c = 5*m+1); c/2^valuation(c, 2); \\ Michel Marcus, Mar 27 2016

a(n) = A000265(A017341(n-1)). - Michel Marcus, Mar 27 2016

cp's OEIS Frontend

A133420 Number of steps to reach 1 under repeated applications of the "5x+1" map of A133419, or -1 if 1 is never reached.

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A133423 Analog of A006684 for the 5x+1 problem (cf. A133419).

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A133424 Analog of A060410 for the 5x+1 problem (cf. A133419).

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A232711 Conjectured list of numbers whose trajectory under the '5x+1' map eventually reaches 1.

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A133421 Image of n under one application of the "7x+1" map.

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A133425 Analog of A006684 for the 7x+1 problem (cf. A133421).

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A133426 Analog of A060410 for the 7x+1 problem (cf. A133421).

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A133422 Number of steps to reach 1 under repeated applications of the "7x+1" map of A133421, or -1 if 1 is never reached.

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A332018 a(n) = A038502(A000265(n)) if n is even or n == 0 (mod 3), a(n) = A038502(A000265(5*n + 1)) otherwise.

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