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%I #50 Jul 01 2022 22:06:26
%S 9,8,8,8,8,8,9,8,8,11,9,9,12,12,8,14,10,13,9,9,9,15,11,17,9,12,9,13,
%T 10,10,10,12,9,10,9,13,9,11,10,12,16,9,12,13,16,9,9,10,9,10,11,11,9,
%U 16,10,11,9,10,10,10,9,10,13,18,9,11,10,9,11,12,13,15,9,12,9,11,13,15,10,9,11,11,11,10,11,11,13,14,10,10,10,10,9,12,10,15,17,10,13,9
%N a(n) is the number of distinct primes produced by starting with the n-th prime p and repeatedly looking at all the prime factors of 2p+1, and then performing the same process (double, add 1, find all prime factors) with those primes; a(n) = -1 if this produces infinitely many primes.
%C Based on a question posed by _James Propp_. Terms computed by _Michael Kleber_.
%C _W. Edwin Clark_ observes (Jun 16 2018) that, based on analysis of the first 10^5 primes, the procedure always ends with {3, 5, 7, 11, 13, 19, 23, 47}, which is sequence A020575. In particular, it appears that the total number of primes obtained is always finite.
%C First occurrences are in A316226.
%D James Propp, Posting to Math Fun Mailing List, Jun 16 2018
%H Hans Havermann, <a href="/A305382/b305382.txt">Table of n, a(n) for n = 1..10000</a>
%e a(1)=9: Starting with the first prime, 2, we see that:
%e 2 -> 5 -> 11 -> 23 -> 47 -> 95=5*19,
%e 19 -> 39=3*13,
%e 3 -> 7 -> 15=3*5,
%e 13 -> 27=3*3*3,
%e which produces 9 different primes, 2 3 5 7 11 13 19 23 47.
%t propp1[p_] := propp1[p] = #[[1]] & /@ FactorInteger[2*p + 1];
%t propp[p_Integer] := propp[{p}];
%t propp[s_List] := propp[s, Union[s, Union @@ propp1 /@ s]];
%t propp[s_, t_] := If[s == t, s, If[Length[t] > 1000, OVERFLOW[t], propp[t]]];
%t Table[Length[propp[Prime[n]]], {n, 100}] (* _Michael Kleber_, Jun 16 2018 *)
%t g[lst_List] := Union@ Join[lst, First@# & /@ Flatten[FactorInteger[2 lst + 1], 1]]; f[n_] := Length@ NestWhile[g@# &, {Prime@ n}, UnsameQ, All]; Table[ f[n], {n, 100}] (* _Robert G. Wilson v_, Jun 17 2018 *)
%o (Python)
%o from sympy import prime, primefactors
%o def a(n):
%o pn = prime(n)
%o reach, expand = {pn}, [pn]
%o while len(expand) > 0:
%o p = expand.pop()
%o for q in primefactors(2*p+1):
%o if q not in reach:
%o expand.append(q)
%o reach.add(q)
%o return len(reach)
%o print([a(n) for n in range(1, 101)]) # _Michael S. Branicky_, Jun 29 2022
%Y Cf. A020575, A306035, A316226.
%K nonn
%O 1,1
%A _N. J. A. Sloane_, Jun 17 2018
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