A057522 -id:A057522 - cp's OEIS frontend

A057534 a(n+1) = a(n)/2 if 2|a(n), a(n)/3 if 3|a(n), a(n)/5 if 5|a(n), a(n)/7 if 7|a(n), a(n)/11 if 11|a(n), a(n)/13 if 13|a(n), otherwise 17*a(n)+1.

61, 1038, 519, 173, 2942, 1471, 25008, 12504, 6252, 3126, 1563, 521, 8858, 4429, 75294, 37647, 12549, 4183, 71112, 35556, 17778, 8889, 2963, 50372, 25186, 12593, 1799, 257, 4370, 2185, 437, 7430, 3715, 743, 12632, 6316, 3158, 1579, 26844, 13422

Offset: 0

Murad A. AlDamen (Divisibility(AT)yahoo.com), Oct 17 2000

This is the '17x+1' map. The 'Px+1 map': if x is divisible by any prime < P then divide out these primes one at a time starting with the smallest; otherwise multiply x by P and add 1.

Sequence has period 84. - Alois P. Heinz, Jan 19 2021

Ray Chandler, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Collatz problem
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

Cf. A057446, A057216 (short version), A057522, A057614.

with(numtheory): a := proc(n) option remember: local k; if n=0 then RETURN(61); fi: for k from 1 to 6 do if a(n-1) mod ithprime(k) = 0 then RETURN(a(n-1)/ithprime(k)); fi: od: RETURN(17*a(n-1)+1) end:

a[n_] := a[n] = Which[n == 0, 61, n <= 84, Module[{k}, For[k = 1, k < PrimePi[17], k++, If[Mod[a[n - 1], Prime[k]] == 0, Return[a[n - 1]/Prime[k]]]]; Return[17*a[n - 1] + 1]], True, a[n - 84]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Aug 22 2023, after Maple code *)

a(n)=if(n, n=a(n-1); if(n%2, if(n%3, if(n%5, if(n%7, if(n%11, if(n%13, 17*n+1, n/13), n/11), n/7), n/5), n/3), n/2), 61) \\ Charles R Greathouse IV, Oct 13 2022

More terms from James Sellers and Larry Reeves (larryr(AT)acm.org), Oct 18 2000

A057614 a(n+1) = a(n)/2 if 2|a(n), a(n)/3 if 3|a(n), a(n)/5 if 5|a(n), a(n)/7 if 7|a(n), otherwise 11*a(n)+1.

Original entry on oeis.org

17, 188, 94, 47, 518, 259, 37, 408, 204, 102, 51, 17, 188, 94, 47, 518, 259, 37, 408, 204, 102, 51, 17, 188, 94, 47, 518, 259, 37, 408, 204, 102, 51, 17, 188, 94, 47, 518, 259, 37, 408, 204, 102, 51, 17, 188, 94, 47, 518, 259, 37, 408, 204, 102, 51, 17, 188, 94, 47, 518, 259, 37, 408, 204, 102, 51

Offset: 0

Murad A. AlDamen (Divisibility(AT)yahoo.com), Oct 17 2000

This is the '11x+1' map. The 'Px+1 map': if x is divisible by any prime < P then divide out these primes one at a time starting with the smallest; otherwise multiply x by P and add 1.

Sequence has period 11. - Alois P. Heinz, Jan 19 2021

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Collatz problem
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,1).

Cf. A057446, A057216, A057522, A057534.

NestList[Which[Mod[#,2]==0,#/2,Mod[#,3]==0,#/3,Mod[#,5]==0,#/5,Mod[#,7]==0,#/7,True,11#+1]&,17,70] (* or *) PadRight[{},100,{17,188,94,47,518,259,37,408,204,102,51}] (* Harvey P. Dale, Jun 30 2025 *)

A057216 To get next term, multiply by 17, add 1 and discard any prime factors < 17.

Original entry on oeis.org

61, 173, 1471, 521, 4429, 4183, 2963, 257, 437, 743, 1579, 2237, 3803, 2309, 19627, 5561, 47269, 14881, 3833, 32581, 263, 43, 61, 173, 1471, 521, 4429, 4183, 2963, 257, 437, 743, 1579, 2237, 3803, 2309, 19627, 5561, 47269, 14881, 3833, 32581, 263, 43

Offset: 0

Murad A. AlDamen (Divisibility(AT)yahoo.com), Oct 17 2000

This is the '17x+1' map. The 'Px+1 map': if x is divisible by any prime < P then divide out these primes one at a time starting with the smallest; otherwise multiply x by P and add 1.

Sequence has period 22. - Alois P. Heinz, Jan 15 2021

			61 -> 17*61+1 = 1038 = 2*3*173 -> 173, so second term is 173.

Eric Weisstein's World of Mathematics, Collatz problem
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A057446, A057522, A057534 (long version), A057614.

a[n_] := a[n] = Which[n == 0, 61, n <= 22, Times @@ Power @@@ Select[ FactorInteger[17 a[n - 1] + 1], #[[1]] >= 17&], True, a[n - 22]];
Table[a[n], {n, 0, 43}] (* Jean-François Alcover, Aug 21 2023 *)

lista(nn) = {my(x=61); for (n=1, nn, print1(x, ", "); my(f=factor(17*x+1)); for (k=1, #f~, if (f[k,1] < 17, f[k,1] = 1)); x = factorback(f););} \\ Michel Marcus, Jan 19 2021

More terms from James Sellers and Larry Reeves (larryr(AT)acm.org), Oct 18 2000

A057446 To get next term, multiply by 13, add 1 and discard any prime factors < 13.

Original entry on oeis.org

73, 19, 31, 101, 73, 19, 31, 101, 73, 19, 31, 101, 73, 19, 31, 101, 73, 19, 31, 101, 73, 19, 31, 101, 73, 19, 31, 101, 73, 19, 31, 101, 73, 19, 31, 101, 73, 19, 31, 101, 73, 19, 31, 101, 73, 19, 31, 101, 73, 19, 31, 101, 73, 19, 31, 101, 73, 19, 31, 101

Offset: 0

Murad A. AlDamen (Divisibility(AT)yahoo.com), Oct 17 2000

This is the '13x+1' map. The 'Px+1 map': if x is divisible by any prime < P then divide out these primes one at a time starting with the smallest; otherwise multiply x by P and add 1.

Sequence has period 4. - Alois P. Heinz, Jan 19 2021

			73 -> 13*73+1 = 950 = 2*5^2*19 -> 19, so second term is 19.

Eric Weisstein's World of Mathematics, Collatz problem
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Cf. A057216, A057522 (long version), A057534, A057614.

m13[n_]:=First[Times@@@Select[FactorInteger[13 n+1],#[[1]]>11&]]; NestList[ m13,73,80] (* or *) PadRight[{},80,{73,19,31,101}] (* Harvey P. Dale, Apr 16 2019 *)
a[n_] := a[n] = Which[n == 0, 73, n <= 4, Times @@ Power @@@ Select[ FactorInteger[13 a[n - 1] + 1], #[[1]] >= 13&], True, a[n - 4]];
Table[a[n], {n, 0, 59}] (* Jean-François Alcover, Aug 21 2023 *)

A057688 Trajectory of 5 under the '5x+1' map.

Original entry on oeis.org

5, 26, 13, 66, 33, 11, 56, 28, 14, 7, 36, 18, 9, 3, 1, 6, 3, 1, 6, 3, 1, 6, 3, 1, 6, 3, 1, 6, 3, 1, 6, 3, 1, 6, 3, 1, 6, 3, 1, 6, 3, 1, 6, 3, 1, 6, 3, 1, 6, 3, 1, 6, 3, 1, 6, 3, 1, 6, 3, 1, 6, 3, 1, 6, 3, 1, 6, 3, 1, 6, 3, 1, 6, 3, 1, 6, 3, 1, 6, 3, 1, 6, 3, 1, 6, 3, 1, 6

Offset: 0

N. J. A. Sloane, Oct 20 2000

The 'Px + 1 map': if x is divisible by any prime less than P then divide out these primes one at a time starting with the smallest; otherwise multiply x by P and add 1. This is similar to A057684, but with P = 5 instead of P = 13. - Alonso del Arte, Jul 04 2015

			7 is odd and not divisible by 3, so it's followed by 5 * 7 + 1 = 36.
36 is even, so it's followed by 36/2 = 18.
18 is even, so it's followed by 18/2 = 9.
9 is odd and divisible by 3, so it's followed by 9/3 = 3.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Cf. A057446, A057216, A057522, A057534, A057614. See also A033478, A057684, A057685, A057686, A057687, A057689, A057690, A057691.

Cf. A259207.

NestList[If[EvenQ[#], #/2, If[Mod[#, 3] == 0, #/3, 5# + 1]] &, 5, 100] (* Alonso del Arte, Jul 04 2015 *)

Vec((5 + 26*x + 13*x^2 + 61*x^3 + 7*x^4 - 2*x^5 - 10*x^6 - 5*x^7 + 3*x^8 - 49*x^9 + 8*x^10 + 4*x^11 + 2*x^12 - 33*x^13 - 17*x^14 - 3*x^15) / ((1 - x)*(1 + x + x^2)) + O(x^100)) \\ Colin Barker, Oct 10 2019

a(0) = 5, a(n) = a(n - 1)/2 if a(n - 1) is even, a(n) = a(n - 1)/3 if a(n - 1) is odd and divisible by 3, a(n) = 5a(n - 1) otherwise.
From Colin Barker, Oct 10 2019: (Start)
G.f.: (5 + 26*x + 13*x^2 + 61*x^3 + 7*x^4 - 2*x^5 - 10*x^6 - 5*x^7 + 3*x^8 - 49*x^9 + 8*x^10 + 4*x^11 + 2*x^12 - 33*x^13 - 17*x^14 - 3*x^15) / ((1 - x)*(1 + x + x^2)).
a(n) = a(n-3) for n>15.
(End)

A057689 Maximal term in trajectory of P under the 'Px+1' map, where P = n-th prime, or -1 if no such term exists.

Original entry on oeis.org

16, 66, 50, 672, 20372, 494, 36918, 1404, 12210, 4248, 5070, 1682, 1850, 2210, 35882, 102720, 94484303672, 30084, 178992, 5330, 246560, 6890, 294253314, 8416400, 515202, 134004, 2810784, 2810883506682183650, 377198408, 320168

Offset: 2

N. J. A. Sloane, Oct 20 2000

See A057684 for definition.

			For n=3, P=7: trajectory of 7 is 7, 50, 25, 5, 1, 8, 4, 2, 1, 8, 4, 2, 1, 8, 4, 2, 1, ..., which has maximal term 50, cycle length 4 and there are 4 terms before it enters the cycle.

Michael S. Branicky, Table of n, a(n) for n = 2..10001 (terms 2..1000 from T. D. Noe)

Cf. A057446, A057216, A057522, A057534, A057614. See also A033478, A057688, A057684, A057685, A057686, A057687, A057690, A057691.

Px1[p_,n_]:=Catch[For[i=1,iPaolo Xausa, Dec 11 2023 *)

from sympy import prime, primerange
def a(n):
    P = prime(n)
    x, plst, seen = P, list(primerange(2, P)), set()
    while x > 1 and x not in seen:
        seen.add(x)
        x = next((x//p for p in plst if x%p == 0), P*x+1)
    return max(seen)
print([a(n) for n in range(2, 32)]) # Michael S. Branicky, Dec 11 2023

More terms from Larry Reeves (larryr(AT)acm.org), Nov 08 2000

A057690 Length of cycle in trajectory of P under the 'Px+1' map, where P = n-th prime, or -1 if trajectory does not cycle.

Original entry on oeis.org

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

Offset: 2

N. J. A. Sloane, Oct 20 2000

See A057684 for definition.

Note that not all cycles for the iteration starting with p contain the number 1; a(60), for the prime 281, is the first example of this. Its iterates are: 281, 78962, 39481, 3037, 853398, 426699, 142233, 47411, 6773, 521, 146402, 73201, 1031, 289712, 144856, 72428, 36214, 18107, 953, 267794, 133897, with the last 12 terms cycling. Another example is provided by 2543, the 372nd prime. - T. D. Noe, Apr 02 2008

			For n=4, P=7: trajectory of 7 is 7, 50, 25, 5, 1, 8, 4, 2, 1, 8, 4, 2, 1, 8, 4, 2, 1, ..., which has maximal term 50, cycle length 4 and there are 4 terms before it enters the cycle.

Michel Marcus, Table of n, a(n) for n = 2..10000

Cf. A057446, A057216, A057522, A057534, A057614. See also A033478, A057688, A057684, A057685, A057686, A057687, A057689, A057691.

Cf. A023514.

Px1[p_,n_]:=Catch[For[i=1,iPaolo Xausa, Dec 11 2023 *)

f(m, p) = {forprime(q=2, precprime(p-1), if (! (m % q), return (m/q));); m*p+1;}
a(n) = {my(p=prime(n), x=p, list = List()); listput(list, x); while (1, x = f(x, p); for (i=1, #list, if (x == list[i], return (#list - i + 1));); listput(list, x););} \\ Michel Marcus, Jan 12 2021

from sympy import prime, primerange
def a(n):
    P = prime(n)
    x, plst, traj, seen = P, list(primerange(2, P)), [], set()
    while x not in seen:
        traj.append(x)
        seen.add(x)
        x = next((x//p for p in plst if x%p == 0), P*x+1)
    return len(traj) - traj.index(x)
print([a(n) for n in range(2, 107)]) # Michael S. Branicky, Dec 11 2023

a(n) = A023514(n)+1 if the cycle contains the number 1. - Jon Maiga, Jan 12 2021

More terms from Larry Reeves (larryr(AT)acm.org), Nov 08 2000

Corrected by T. D. Noe, Apr 02 2008

A057691 Number of terms before entering cycle in trajectory of P under the 'Px+1' map, where P = n-th prime, or -1 if trajectory does not cycle.

Original entry on oeis.org

5, 13, 4, 10, 25, 11, 68, 14, 39, 34, 9, 4, 5, 5, 16, 16, 234, 23, 16, 5, 11, 5, 63, 116, 18, 18, 33, 288, 47, 29, 317, 14, 12, 61, 60, 6, 16, 10, 5, 14, 46, 5, 6, 15, 105, 4, 11, 48, 44, 5, 6, 10, 5, 55, 15, 14, 25, 17, 9, 16, 6, 7, 26, 5, 33, 46, 5, 16, 23, 13, 15, 11, 16, 14, 11

Offset: 2

N. J. A. Sloane, Oct 20 2000

See A057684 for definition.

			For n=3, P=7: trajectory of 7 is 7, 50, 25, 5, 1, 8, 4, 2, 1, 8, 4, 2, 1, 8, 4, 2, 1, ..., which has maximal term 50, cycle length 4 and there are 4 terms before it enters the cycle.

Michael S. Branicky, Table of n, a(n) for n = 2..10001 (terms 2..1000 from T. D. Noe)

Cf. A057446, A057216, A057522, A057534, A057614. See also A033478, A057688, A057684, A057685, A057686, A057687, A057689, A057690.

Px1[p_,n_]:=Catch[For[i=1,iPaolo Xausa, Dec 11 2023 *)

from sympy import prime, primerange
def a(n):
    P = prime(n)
    x, plst, traj, seen = P, list(primerange(2, P)), [], set()
    while x not in seen:
        traj.append(x)
        seen.add(x)
        x = next((x//p for p in plst if x%p == 0), P*x+1)
    return traj.index(x)
print([a(n) for n in range(2, 82)]) # Michael S. Branicky, Dec 11 2023

More terms from Larry Reeves (larryr(AT)acm.org), Nov 08 2000

A057684 Trajectory of 13 under the '13x+1' map.

Original entry on oeis.org

13, 170, 85, 17, 222, 111, 37, 482, 241, 3134, 1567, 20372, 10186, 5093, 463, 6020, 3010, 1505, 301, 43, 560, 280, 140, 70, 35, 7, 1, 14, 7, 1, 14, 7, 1, 14, 7, 1, 14, 7, 1, 14, 7, 1, 14, 7, 1, 14, 7, 1, 14, 7, 1, 14, 7, 1, 14, 7, 1, 14, 7, 1, 14, 7

Offset: 0

N. J. A. Sloane, Oct 20 2000

The 'Px+1 map': if x is divisible by any prime < P then divide out these primes one at a time starting with the smallest; otherwise multiply x by P and add 1.

Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Cf. A057446, A057216, A057522, A057534, A057614. See also A033478, A057688, A057685, A057686, A057687, A057689, A057690, A057691.

with(numtheory): a := proc(n,S,Q) option remember: local k; if n=0 then RETURN(S); fi: for k from 1 to Q do if a(n-1,S,Q) mod ithprime(k) = 0 then RETURN(a(n-1,S,Q)/ithprime(k)); fi: od: RETURN(ithprime(Q+1)*a(n-1,S,Q)+1) end; # run with S=13 and Q=5.

a[n_, S_, Q_] := a[n, S, Q] = Module[{k}, If[n == 0, S, For[k = 1, k <= Q, k++, If[Mod[a[n-1, S, Q], Prime[k]] == 0, Return[a[n-1, S, Q]/Prime[k]]] ]; Prime[Q+1]*a[n-1, S, Q] + 1]];
Table[a[n, 13, 5], {n, 0, 60}] (* Jean-François Alcover, Jul 13 2016, adapted from Maple *)

A057685 Trajectory of 19 under the `19x+1' map.

Original entry on oeis.org

19, 362, 181, 3440, 1720, 860, 430, 215, 43, 818, 409, 7772, 3886, 1943, 36918, 18459, 6153, 2051, 293, 5568, 2784, 1392, 696, 348, 174, 87, 29, 552, 276, 138, 69, 23, 438, 219, 73, 1388, 694, 347, 6594, 3297, 1099, 157, 2984, 1492, 746

Offset: 0

N. J. A. Sloane, Oct 20 2000

See A057684 for definition.

T. D. Noe, Table of n, a(n) for n=0..72
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Cf. A057446, A057216, A057522, A057534, A057614. See also A033478, A057688, A057684, A057685, A057686, A057687, A057689, A057690, A057691.

Px1[p_,n_]:=Catch[For[i=1,iPaolo Xausa, Dec 10 2023 *)

cp's OEIS Frontend

A057534 a(n+1) = a(n)/2 if 2|a(n), a(n)/3 if 3|a(n), a(n)/5 if 5|a(n), a(n)/7 if 7|a(n), a(n)/11 if 11|a(n), a(n)/13 if 13|a(n), otherwise 17*a(n)+1.

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A057614 a(n+1) = a(n)/2 if 2|a(n), a(n)/3 if 3|a(n), a(n)/5 if 5|a(n), a(n)/7 if 7|a(n), otherwise 11*a(n)+1.

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A057216 To get next term, multiply by 17, add 1 and discard any prime factors < 17.

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A057446 To get next term, multiply by 13, add 1 and discard any prime factors < 13.

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A057688 Trajectory of 5 under the '5x+1' map.

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A057689 Maximal term in trajectory of P under the 'Px+1' map, where P = n-th prime, or -1 if no such term exists.

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A057690 Length of cycle in trajectory of P under the 'Px+1' map, where P = n-th prime, or -1 if trajectory does not cycle.

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