A177331 -id:A177331 - cp's OEIS frontend

A177330 Least k>0 such that (p*2^k-1)/3 is prime, or zero if no k exists, where p=prime(n).

3, 0, 1, 4, 1, 2, 1, 4, 3, 1, 2, 4, 3, 4, 1, 9107, 3, 6, 2, 1, 2, 4, 7, 1, 6, 1, 2, 1, 32, 11, 4, 3, 45, 24, 3, 6, 8, 16, 21, 3, 29, 2, 1, 2, 1, 4, 2, 66, 1, 8, 7, 5, 10, 1, 5, 3, 1, 14, 18, 13, 6, 59, 2, 3, 4, 1, 18, 2, 5, 4, 3, 1, 6, 5016, 8, 3, 15, 14, 3, 12, 3, 46, 5, 2, 4, 3, 5, 4, 1, 2, 1, 3

Offset: 1

T. D. Noe, May 08 2010

nonn

When a(n) is not zero, a(n) is even if p=1 (mod 6); a(n) is odd if p=5 (mod 6). If we let q=(p*2^k-1)/3 be a prime generated by p for some k>0, then the first prime number after q in the Collatz iteration of q is p. When k=1, q is less than p. The primes, other than 3, for which a(n)=0 are in A177331.

Cf. A177000

Table[p=Prime[n]; If[p==3, k=0, k=1; While[q=(p*2^k-1)/3; k<10000 && !PrimeQ[q], k++ ]]; k, {n,100}]

A225424 Least prime p such that prime(n) is the next prime number in the Collatz (3x+1) iteration of p, or 0 if there is no such prime.

Original entry on oeis.org

5, 0, 3, 37, 7, 17, 11, 101, 61, 19, 41, 197, 109, 229, 31

Offset: 1

T. D. Noe, May 22 2013

nonn

The next term, surprisingly, is the 2743-digit (53 * 2^9107 - 1)/3. See A177331 for additional prime numbers, besides 3, that are not in the Collatz iteration of any prime number.

			a(4) = 37 because the Collatz iteration of 37 is {37, 112, 56, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1}, which shows that 7 is the next prime after 37.

Cf. A070165, A177330, A177331.

Table[p = Prime[n]; If[p == 3, q = 0, k = 1; While[q = (p*2^k - 1)/3; k < 10000 && ! PrimeQ[q], k++]]; q, {n, 15}]

cp's OEIS Frontend

A177330 Least k>0 such that (p*2^k-1)/3 is prime, or zero if no k exists, where p=prime(n).

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