A087272 -id:A087272 - cp's OEIS frontend

A055510 Largest odd prime in sequence obtained in 3x+1 (or Collatz) problem starting at n, or 0 if no such prime is found.

0, 0, 5, 0, 5, 5, 17, 0, 17, 5, 17, 5, 13, 17, 53, 0, 17, 17, 29, 5, 0, 17, 53, 5, 29, 13, 1619, 17, 29, 53, 1619, 0, 29, 17, 53, 17, 37, 29, 101, 5, 1619, 0, 43, 17, 17, 53, 1619, 5, 37, 29, 29, 13, 53, 1619, 1619, 17, 43, 29, 101, 53, 61, 1619, 1619, 0, 37, 29, 101, 17, 13

Offset: 1

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

nonn

Eric Weisstein's World of Mathematics, Collatz Problem
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A055509, A087272.

op[n_]:=Module[{lst=NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#!=1&],m}, m= Max[Select[ lst,PrimeQ]];If[m<3,0,m]]; Array[op,70] (* Harvey P. Dale, Apr 30 2011 *)

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

A087274 Prime index of the largest prime factor of 3*prime(n)+1.

Original entry on oeis.org

4, 3, 1, 5, 7, 3, 6, 10, 4, 5, 15, 4, 11, 6, 20, 3, 24, 9, 26, 28, 5, 7, 3, 19, 21, 8, 11, 9, 13, 7, 43, 45, 27, 8, 4, 49, 17, 4, 54, 6, 57, 7, 13, 10, 12, 9, 66, 19, 11, 14, 4, 72, 42, 10, 44, 22, 26, 12, 6, 47, 7, 5, 89, 91, 15, 7, 20, 9, 98, 32, 16, 5, 10, 4, 104, 9, 21, 35, 14, 63, 12, 22

Offset: 1

Labos Elemer, Sep 18 2003

nonn

			n=10: prime(10)=29, max-p-factor(88)=11, pi(11)=5=a(10)<n;
n=11: prime(11)=31, max-p-factor(94)=47, pi(47)=15=a(11)>n;

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000720, A006530, A087272, A087273.

ffi[x_] := Flatten[FactorInteger[x]]; ma[x_] := Part[Reverse[ffi[x]], 2]; Table[PrimePi[ma[3*Prime[w]+1]], {w, 1, 100}]

a(n) = primepi(vecmax(factor(3*prime(n)+1)[, 1])); \\ Michel Marcus, Mar 27 2020

a(n) = A000720(A006530(1+3*A000040(n))).

a(n) = A000720(A087273(n)). - Amiram Eldar, Jul 12 2024

A087963 Exponent of highest power of 2 dividing 3*prime(n)+1.

Original entry on oeis.org

0, 1, 4, 1, 1, 3, 2, 1, 1, 3, 1, 4, 2, 1, 1, 5, 1, 3, 1, 1, 2, 1, 1, 2, 2, 4, 1, 1, 3, 2, 1, 1, 2, 1, 6, 1, 3, 1, 1, 3, 1, 5, 1, 2, 4, 1, 1, 1, 1, 4, 2, 1, 2, 1, 2, 1, 3, 1, 6, 2, 1, 4, 1, 1, 2, 3, 1, 2, 1, 3, 2, 1, 1, 5, 1, 1, 4, 3, 2, 2, 1, 4, 1, 2, 1, 1, 2, 2, 3, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 3, 1, 3, 1, 2, 1

Offset: 1

Labos Elemer, Sep 18 2003

nonn

			For n = 10: p = prime(10) = 29, 3*p + 1 = 88 = 2^3 * 11, a(10) = 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A087272, A087273, A087274, A007814, A087230.

[Valuation(3*NthPrime(n)+1, 2): n in [1..80]]; // Vincenzo Librandi, Sep 01 2016

ffi[x_] := Flatten[FactorInteger[x]]; e2[x_] := Part[[ffi[x]], 2]; Table[e2[3*Prime[w]+1], {w, 1, 100}]
IntegerExponent[3 * Prime[Range[100]] + 1, 2] (* Amiram Eldar, Jul 12 2024 *)

a(n) = valuation(3*prime(n)+1, 2); \\ Michel Marcus, Sep 01 2016

from sympy import prime
def A087963(n): return (~(m:=prime(n)*3+1)&m-1).bit_length() # Chai Wah Wu, Jul 10 2022

a(n) = A007814(3*prime(n)+1).

a(1)=0 corrected by Michel Marcus, Sep 01 2016

A320028 a(n) is the first prime encountered when running the Collatz algorithm (halving and tripling steps) on the number n.

Original entry on oeis.org

2, 3, 2, 5, 3, 7, 2, 7, 5, 11, 3, 13, 7, 23, 2, 17, 7, 19, 5, 2, 11, 23, 3, 19, 13, 41, 7, 29, 23, 31, 2, 19, 17, 53, 7, 37, 19, 59, 5, 41, 2, 43, 11, 17, 23, 47, 3, 37, 19, 29, 13, 53, 41, 83, 7, 43, 29, 59, 23, 61, 31, 137, 2, 37, 19, 67, 17, 13, 53, 71, 7, 73, 37, 113, 19, 29, 59, 79, 5, 61, 41, 83, 2, 2, 43, 131

Offset: 2

Alessandro Polcini, Oct 03 2018

nonn

A modified version of the halving and tripling Collatz algorithm, which stops as soon as the starting number becomes a prime (instead of stopping when the starting number reaches 1).

The plot of this sequence "completes" or "fills" the lower (empty) part of plot of A270570 and evolves in a similar fashion.

			a(4) is 2 because 4/2 = 2 and 2 is prime.
a(6) is 3 because 6/2 = 3 and 3 is prime.
a(15) is 23 because 15*3 + 1 = 46; 46/2 = 23 and 23 is prime.
a(18) is 7 because 18/2 = 9; 9*3 + 1 = 28; 28/2 = 14; 14/2 = 7 and 7 is prime.

Alessandro Polcini, Table of n, a(n) for n = 2..10000 (a(2024) corrected by Michel Marcus, Jun 14 2022)
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A087272, A270570.

int collatzPrime(int i) {
    while(!BigInteger.valueOf(i).isProbablePrime(10) && i > 1) {
        if(i % 2 == 0)
            i /= 2;
        else
            i = 3 * i + 1;
    }
    return i;
}

Array[NestWhile[If[EvenQ@ #, #/2, 3 # + 1] &, #, ! PrimeQ@ # &] &, 86, 2] (* Michael De Vlieger, Nov 07 2018 *)

a(n) = {while (!isprime(n), if (n % 2, n = 3*n+1, n = n/2);); n;} \\ Michel Marcus, Oct 28 2018

a(n) <= A087272(n). - Rémy Sigrist, Oct 08 2018

A087964 a(n) is the least prime p such that exponent of highest power of 2 dividing 3p+1 equals n.

Original entry on oeis.org

3, 17, 13, 5, 53, 149, 1237, 1109, 853, 2389, 3413, 17749, 128341, 70997, 251221, 415061, 218453, 2708821, 27088213, 29709653, 3495253, 85284181, 13981013, 39146837, 794121557, 1498764629, 492131669, 626349397, 13779686741

Offset: 1

Labos Elemer, Sep 18 2003

nonn

			p = 218453 is the first prime so that 3*p+1 = 655360 = (2^18)*5 has 18 as exponent of 2 in 3p+1, thus a(18) = 218453.

Giovanni Resta, Table of n, a(n) for n = 1..1000

Cf. A087272, A087273, A087274, A007814, A087230, A087963.

f:= proc(n)
   local m,t,p;
   t:= 2^n;
   for m from 1 + 4*(n mod 2) by 6 do
     p:= (t*m-1)/3;
     if isprime(p) then return p fi
   od
end proc:
map(f, [$1..100]); # Robert Israel, Nov 18 2017

a[n_] := Module[{m, t = 2^n, p}, For[m = 1 + 4 Mod[n, 2], True, m += 6, p = (t m - 1)/3; If[PrimeQ[p], Return[p]]]];
Array[a, 100] (* Jean-François Alcover, Aug 28 2020, after Robert Israel *)

a(n) = A000040(Min{x; A007814(1 + 3*A000040(x)) = n}).

More terms from Ray Chandler, Sep 21 2003

A299963 a(n) = greatest prime factor of the terms in the Collatz sequence starting at n; a(1) = 1.

Original entry on oeis.org

1, 2, 5, 2, 5, 5, 17, 2, 17, 5, 17, 5, 13, 17, 53, 2, 17, 17, 29, 5, 7, 17, 53, 5, 29, 13, 1619, 17, 29, 53, 1619, 2, 29, 17, 53, 17, 37, 29, 101, 5, 1619, 7, 43, 17, 17, 53, 1619, 5, 37, 29, 29, 13, 53, 1619, 1619, 17, 43, 29, 101, 53, 61, 1619, 1619, 2, 37

Offset: 1

Rémy Sigrist, Feb 22 2018

nonn

The value 3 cannot appear in this sequence.
The value 1619 appears 1654 times among the first 10000 terms; this is visible as a dashed horizontal line in the corresponding scatterplot.
The most frequent values among the first 10000000 terms are:
Value     Number of occurrences among the first 10000000 terms
  -------   ---------------------------------------------------
   283763   16934
  2017817   15701
     1619   15274
    55667   14706
  2717873    9913

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Ordinal transform of the first 10000000 terms
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006530, A055510, A087272, A178168.

Table[Max[FactorInteger[#][[-1,1]]&/@NestWhileList[If[EvenQ[#],#/2,3#+1]&, n,#>1&]], {n,70}] (* Harvey P. Dale, Jun 22 2020 *)

a(n) = my (g=1); while (n>1, my (f=factor(n)); g=max(g,f[#f~,1]); n=if (n%2, 3*n+1, n/2)); return (g)

a(n) = A006530(A178168(n)).
a(2*n) = a(n) for any n > 1.
a(2^k) = 2 for any k > 0.

cp's OEIS Frontend

A055510 Largest odd prime in sequence obtained in 3x+1 (or Collatz) problem starting at n, or 0 if no such prime is found.

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A087274 Prime index of the largest prime factor of 3*prime(n)+1.

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A087963 Exponent of highest power of 2 dividing 3*prime(n)+1.

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A320028 a(n) is the first prime encountered when running the Collatz algorithm (halving and tripling steps) on the number n.

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A087964 a(n) is the least prime p such that exponent of highest power of 2 dividing 3p+1 equals n.

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A299963 a(n) = greatest prime factor of the terms in the Collatz sequence starting at n; a(1) = 1.

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