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A127928 Pure hailstone primes.

%I A127928 #15 Jan 05 2025 19:51:38
%S A127928 3,7,19,37,43,73,79,97,109,127,151,163,181,199,223,241,271,277,307,
%T A127928 313,331,349,367,379,397,421,439,457,487,523,541,547,601,613,619,631,
%U A127928 673,691,709,727,757,811,829,853,883,907,937,997,1009,1033,1051,1069,1087,1117
%N A127928 Pure hailstone primes.
%C A127928 In other words, pure hailstone numbers that are also primes (primes in A061641).
%C A127928 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).
%C A127928 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.
%H A127928 Amiram Eldar, <a href="/A127928/b127928.txt">Table of n, a(n) for n = 1..10000</a>
%H A127928 Douglas J. Shaw, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/44-3/quartshaw03_2006.pdf">The Pure Numbers Generated by the Collatz Sequence</a>, The Fibonacci Quarterly, Vol. 44, Number 3, August 2006, pp. 194-201.
%e A127928 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).
%Y A127928 Cf. A127929, A127930, A061641, A127633, A066903.
%K A127928 nonn
%O A127928 1,1
%A A127928 _Gary W. Adamson_, Feb 07 2007
%E A127928 More terms from _Amiram Eldar_, Feb 28 2020