A127928 Pure hailstone primes.

3, 7, 19, 37, 43, 73, 79, 97, 109, 127, 151, 163, 181, 199, 223, 241, 271, 277, 307, 313, 331, 349, 367, 379, 397, 421, 439, 457, 487, 523, 541, 547, 601, 613, 619, 631, 673, 691, 709, 727, 757, 811, 829, 853, 883, 907, 937, 997, 1009, 1033, 1051, 1069, 1087, 1117

Offset

1,1

Comments

In other words, pure hailstone numbers that are also primes (primes in A061641).

Impure hailstone numbers occur in the trajectories of smaller numbers, using the definition C(n) = (3n+1, n odd; n/2 if n is even). The set of pure hailstone numbers and the subset of pure, prime hailstone numbers; may be obtained through a process of elimination. The rules [cf. Shaw, p. 199] for A127928(n>1) force the terms to be == 1 or 7 mod 18; but not all primes mod 1 or 7 are in A127928. (e.g. 61 == 7 mod 18 and is prime but is not a pure hailstone number).

Shaw, p. 199: If n == 0, 3, 6, 9, 12 or 15 mod 18, then n is pure, but only 3 is prime. If n == 2, 4, 5, 8, 10, 11, 13, 14, 16 or 17 mod 18, then n is impure. If n == 1 or 7 mod 18, then n may be pure or impure.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Douglas J. Shaw, The Pure Numbers Generated by the Collatz Sequence, The Fibonacci Quarterly, Vol. 44, Number 3, August 2006, pp. 194-201.

Example

3 is a pure hailstone (Collatz) number since it does not appear in the orbit of 1 or 2, but 5 is impure since the iterative trajectory of 3 = (10, 5, 16, 8, 4, 2, 1).

Crossrefs

Cf. A127929, A127930, A061641, A127633, A066903.

Sequence in context: A268065 A049490 A112391 * A298125 A047025 A222465

Adjacent sequences: A127925 A127926 A127927 * A127929 A127930 A127931

Keyword

nonn

Author

Gary W. Adamson, Feb 07 2007

Extensions

More terms from Amiram Eldar, Feb 28 2020

Status

approved