

A305382


a(n) is the number of distinct primes produced by starting with the nth 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.


4



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, 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, 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
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OFFSET

1,1


COMMENTS

Based on a question posed by James Propp. Terms computed by Michael Kleber.
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.
First occurrences are in A316226.


REFERENCES

James Propp, Posting to Math Fun Mailing List, Jun 16 2018


LINKS

Hans Havermann, Table of n, a(n) for n = 1..10000


EXAMPLE

a(1)=9: Starting with the first prime, 2, we see that:
2 > 5 > 11 > 23 > 47 > 95=5*19,
19 > 39=3*13,
3 > 7 > 15=3*5,
13 > 27=3*3*3,
which produces 9 different primes, 2 3 5 7 11 13 19 23 47.


MATHEMATICA

propp1[p_] := propp1[p] = #[[1]] & /@ FactorInteger[2*p + 1];
propp[p_Integer] := propp[{p}];
propp[s_List] := propp[s, Union[s, Union @@ propp1 /@ s]];
propp[s_, t_] := If[s == t, s, If[Length[t] > 1000, OVERFLOW[t], propp[t]]];
Table[Length[propp[Prime[n]]], {n, 100}] (* Michael Kleber, Jun 16 2018 *)
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 *)


CROSSREFS

Cf. A020575, A306035, A316226.
Sequence in context: A258752 A195477 A157680 * A011228 A175572 A263984
Adjacent sequences: A305379 A305380 A305381 * A305383 A305384 A305385


KEYWORD

nonn


AUTHOR

N. J. A. Sloane, Jun 17 2018


STATUS

approved



