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%I #21 Sep 07 2018 03:41:31
%S 149,173,307,373,439,443,541,557,563,617,827,863,1297,1303,1373,1453,
%T 1489,1627,1657,1667,1733,1783,1861,1901,2029,2053,2393,2423,2591,
%U 2609,2647,2657,2677,2767,3037,3067,3253,3319,3343,3361,3433,3461,3467,3517,3659
%N 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.
%e 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.
%o (Python)
%o def length_collatz_chain(start):
%o i=0
%o while start != 1:
%o if (start % 2 == 0):
%o start = start / 2
%o else:
%o start = 3 * start + 1
%o i = i+1
%o return i
%o def is_prime(num):
%o if num == 1: return(0)
%o for k in range(2, num):
%o if (num % k) == 0:
%o return(0)
%o return(1)
%o collatz = []
%o nmax=10000
%o for i in range(nmax):
%o collatz.append(0)
%o collatz.append(0)
%o for i in range(nmax):
%o start=i+1
%o collatz[start]=length_collatz_chain(start)
%o lista_elem=[]
%o elem=[]
%o for i in range(1,nmax):
%o if is_prime(collatz[i]) and is_prime(i):
%o elem.append(i)
%o elem.append(collatz[i])
%o lista_elem.append(elem)
%o elem=[]
%o result=""
%o for i in range(len(lista_elem)-1):
%o if lista_elem[i][1]==lista_elem[i+1][1]:
%o result=result+str(lista_elem[i][0])+","
%o print(result)
%o (PARI) nbs(n) = my(s); while(n>1, n=if(n%2, 3*n+1, n/2); s++); s; \\ A006577
%o 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
%Y Cf. A006577.
%Y Subsequence of A176112.
%K nonn
%O 1,1
%A _Pierandrea Formusa_, Jul 07 2018
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