1,1

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.

(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

Cf. A006577.

Subsequence of A176112.

Sequence in context: A190654 A308895 A100723 * A178127 A307472 A209619

Adjacent sequences: A316586 A316587 A316588 * A316590 A316591 A316592

Pierandrea Formusa, Jul 07 2018

approved