A390817 Number of tripling steps to reach 1 in the 3x+1 (Collatz) problem starting with the n-th Mersenne prime.

2, 5, 39, 15, 56, 80, 61, 162, 309, 528, 516, 593, 2417, 3051, 6118, 10684, 11007, 15575, 19982, 21777, 46391, 48126, 55046, 95136, 105015, 111783, 214150, 414951, 530772, 634078, 1040668, 3651984, 4142867, 6062053, 6734691, 14344249, 14561774, 33581089, 64891802

Offset

1,1

Comments

The n-th Mersenne prime is A000668(n) = 2^A000043(n) - 1, and its total number of steps is A181777(n).

Links

V. Barbera, Table of n, a(n) for n = 1..50

Formula

a(n) = A006667(A000668(n)).

a(n) = A390816(A000043(n)).

a(n) = A075680(2^(A000043(n) - 1)).

a(n) = A075680(A379724(n)/2 + 1).

Example

For n=2, the second Mersenne prime is A000668(2) = 7 and there ate a(2) = 5 tripling steps on its way to 1.

Mathematica

a[n_] := -1 + Count[NestWhileList[If[OddQ[#], 3*# + 1, #/2] &, 2^MersennePrimeExponent[n] - 1, # > 1 &], _?OddQ]; Array[a, 20] (* Amiram Eldar, Nov 21 2025 *)

Prog

(Python)
# Exponents p with 2^p - 1 prime (A000043), for p <= 127
mersenne_prime_exponents = [2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127]
def collatz_tripling_steps(n: int) -> int: # A006667(n)
    steps, x = 0, n
    while x != 1:
        if x % 2 == 0:
            x //= 2
        else:
            x = 3*x + 1
            steps += 1
    return steps
print([collatz_tripling_steps((1 << p) - 1) for p in mersenne_prime_exponents])

Crossrefs

Cf. A000043, A000225, A000668, A006667, A075680, A181777.

Subsequence of A390816.

Sequence in context: A228837 A206155 A135378 * A077398 A255550 A362800

Adjacent sequences: A390814 A390815 A390816 * A390818 A390819 A390820

Keyword

nonn

Author

Stephen R. Campbell, Nov 20 2025

Extensions

a(18)-a(30) from Amiram Eldar, Nov 21 2025

a(31)-a(35) from Sean A. Irvine, Nov 29 2025

a(36)-a(39) from Jinyuan Wang, Dec 07 2025

Status

approved