login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A316589 Prime numbers p whose number of steps to reach 1 in Collatz (3x+1) problem is a prime number k and, in addition, the least prime number greater than p that also reaches 1 in the same problem in a prime number of steps also does so in k steps. 0
149, 173, 307, 373, 439, 443, 541, 557, 563, 617, 827, 863, 1297, 1303, 1373, 1453, 1489, 1627, 1657, 1667, 1733, 1783, 1861, 1901, 2029, 2053, 2393, 2423, 2591, 2609, 2647, 2657, 2677, 2767, 3037, 3067, 3253, 3319, 3343, 3361, 3433, 3461, 3467, 3517, 3659 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

LINKS

Table of n, a(n) for n=1..45.

EXAMPLE

149 belongs to this sequence as it is prime, it satisfies the Collatz conjecture in 23 (that is prime) steps; no other prime number greater than 149 and less than 163 satisfies the conjecture in a prime number of steps (151 does it in 15 steps; 157 in 36 steps); and the prime number 163 also satisfies it in 23 steps, just as 149 does.

PROG

(Python)

def length_collatz_chain(start):

    i=0

    while start != 1:

        if (start % 2 == 0):

            start = start / 2

        else:

            start = 3 * start + 1

        i = i+1

    return i

def is_prime(num):

    if num == 1: return(0)

    for k in range(2, num):

       if (num % k) == 0:

           return(0)

    return(1)

collatz = []

nmax=10000

for i in range(nmax):

    collatz.append(0)

collatz.append(0)

for i in range(nmax):

    start=i+1

    collatz[start]=length_collatz_chain(start)

lista_elem=[]

elem=[]

for i in range(1, nmax):

    if is_prime(collatz[i]) and is_prime(i):

        elem.append(i)

        elem.append(collatz[i])

        lista_elem.append(elem)

        elem=[]

result=""

for i in range(len(lista_elem)-1):

    if lista_elem[i][1]==lista_elem[i+1][1]:

        result=result+str(lista_elem[i][0])+", "

print(result)

(PARI) nbs(n) = my(s); while(n>1, n=if(n%2, 3*n+1, n/2); s++); s; \\ A006577

lista(nn) = {vp = primes(nn); vs = select(x->isprime(nbs(x)), vp, 1); vpok = vector(#vs, k, prime(vs[k])); vpoks = vector(#vpok, k, nbs(vpok[k])); for (i=1, #vpoks-1, if (vpoks[i] == vpoks[i+1], print1(vpok[i], ", ")); ); } \\ Michel Marcus, Jul 27 2018

CROSSREFS

Cf. A006577.

Subsequence of A176112.

Sequence in context: A190654 A308895 A100723 * A178127 A307472 A209619

Adjacent sequences:  A316586 A316587 A316588 * A316590 A316591 A316592

KEYWORD

nonn

AUTHOR

Pierandrea Formusa, Jul 07 2018

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 21 14:54 EDT 2020. Contains 337272 sequences. (Running on oeis4.)