A387787 Positive integers k for which the Collatz 3x+1 dropping value (first value < k) is prime.

3, 4, 6, 7, 9, 10, 14, 17, 19, 22, 25, 26, 27, 31, 34, 38, 41, 43, 46, 49, 51, 55, 57, 58, 62, 63, 71, 74, 81, 82, 83, 86, 89, 91, 94, 97, 103, 105, 106, 111, 118, 119, 122, 129, 134, 137, 142, 145, 146, 147, 158, 166, 169, 175, 178, 179, 185, 194, 201, 202

Offset

1,1

Comments

All k = 2*prime (A100484) are terms so the sequence is infinite.

A given prime can be the drop value of various different k, as for example 23 is the drop value of k = 27, 31, 46.

Links

Table of n, a(n) for n=1..60.

Example

For k = 3: 3 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1. The first term < 3 is 2, which is prime, so 3 is a term.
For k = 8: 8 -> 4 -> 2 -> 1. The first term < 8 is 4, which is composite, so 8 is not a term.

Prog

(Python)
from sympy import isprime
def T(n):
    return n//2 if n % 2 == 0 else 3*n + 1
def drop(k):
    x = k
    while x >= k:
        x = T(x)
    return x
def isok(k):
    return isprime(drop(k))
(PARI) isok(k) = if (k>1, my(m=k); while(m >= k, m = if (m%2, 3*m+1, m/2)); isprime(m)); \\ Michel Marcus, Dec 06 2025

Crossrefs

Cf. A100484, A060565, A070165, A006370, A074473.

Sequence in context: A190858 A026315 A143264 * A126332 A039026 A123540

Adjacent sequences: A387784 A387785 A387786 * A387788 A387789 A387790

Keyword

nonn,easy

Author

Aied Sulaiman, Dec 04 2025

Status

approved