A087273 -id:A087273 - cp's OEIS frontend

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

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

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

A276827 Primes p such that the greatest prime factor of 3*p+1 is at most 5.

Original entry on oeis.org

3, 5, 13, 53, 83, 853, 2083, 3413, 5333, 85333, 208333, 218453, 341333, 3495253, 5461333, 8533333, 13981013, 83333333, 853333333, 22369621333, 218453333333, 341333333333, 2236962133333, 3665038759253, 53333333333333, 91625968981333, 203450520833333, 1333333333333333

Offset: 1

Robert Israel, Sep 19 2016

nonn

Prime(i) such that A087273(i) <= 5.

Robert Israel, Table of n, a(n) for n = 1..1645

Cf. A087273.

Contains A093671, A093674, and A093676.

N = 10^20: # to get all terms <= N
Res:= {}:
for a from 0 to ilog2(floor((3*N+1)/5)) do
  twoa:= 2^a;
  for b from (a mod 2) by 2 do
    p:= (twoa*5^b-1)/3;
    if p > N then break fi;
    if isprime(p) then
      Res:= Res union {p};
    fi
od od:
sort(convert(Res,list));

Select[Prime@ Range[10^6], FactorInteger[3 # + 1][[-1, 1]] <= 5 &] (* Michael De Vlieger, Sep 19 2016 *)

list(lim)=my(v=List(),s,t); lim=lim\1*3 + 1; for(i=0,logint(lim\2,5), t=if(i%2,2,4)*5^i; while(t<=lim, if(isprime(p=t\3), listput(v,p)); t<<=2)); Set(v) \\ Charles R Greathouse IV, Sep 19 2016

A276357 Primes of the form (p*2^x-1)/3, where p is also prime and x is a positive integer.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 29, 31, 37, 41, 47, 53, 59, 61, 67, 71, 89, 97, 101, 109, 127, 131, 137, 149, 151, 157, 167, 179, 181, 197, 211, 229, 239, 241, 257, 269, 277, 281, 307, 311, 347, 349, 379, 389, 397, 409, 421, 431, 439, 449, 461, 467, 479, 509, 547, 571, 577, 587

Offset: 1

Michael Cader Nelson, Aug 31 2016

nonn

Relationship to Collatz (3x+1) problem: when one of these primes appears in a hailstone sequence, the next odd number in the sequence must be prime. - Michael Cader Nelson, Jul 03 2020

			3 is in the sequence because 3 = (5*2^1-1)/3 and both 3 and 5 are prime numbers; while 23 is not in the sequence because the only positive integer values (p,x) to give 23 are (35,1) and 35 is not prime.

Cf. A087273, A087963. A177330 (lists all exponents x).

mx = 590; Select[ Sort@ Flatten@ Table[(Prime[p]*2^x - 1)/3, {x, Log2[mx/3]}, {p, PrimePi[3 mx/2^x]}], PrimeQ] (* Robert G. Wilson v, Nov 01 2016 *)

lista(nn) = {forprime(p=2, nn, z = 3*p+1; x = valuation(z, 2); for (ex = 1, x, if (isprime(z/2^ex), print1(p, ", "); break;);););} \\ Michel Marcus, Sep 01 2016

The value of p is (3*a(n)+1)/2^x as well as the respective term in A087273 evaluated for a(n), while the value of x is the related exponent in A087963 unless 3*a(n)+1 is a power of 2 (e.g., n = 1).

Corrected and extended by Michel Marcus, Sep 01 2016

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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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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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A276827 Primes p such that the greatest prime factor of 3*p+1 is at most 5.

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Maple

Mathematica

PARI

A276357 Primes of the form (p*2^x-1)/3, where p is also prime and x is a positive integer.

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Author

Keywords

Comments

Examples

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Crossrefs

Programs

Mathematica

PARI

Formula

Extensions