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.

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

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