

A298904


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 2p1, and then performing the same process (double, subtract 1, find all prime factors) with those primes; a(n) = 1 if this produces infinitely many primes.


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3, 2, 2, 4, 5, 3, 6, 6, 3, 6, 7, 6, 3, 7, 8, 5, 4, 6, 9, 9, 6, 5, 6, 5, 7, 10, 4, 10, 8, 3, 7, 7, 5, 9, 6, 8, 4, 4, 7, 4, 7, 7, 8, 6, 8, 8, 8, 6, 9, 8, 8, 6, 8, 8, 7, 5, 8, 11, 8, 7, 4, 4, 6, 4, 3, 9, 16, 10, 6, 8, 9, 7, 6, 7, 10, 7, 9, 7, 6, 12, 8, 7, 6, 5, 9, 5, 5, 7, 7, 7, 5, 10, 10, 9, 8, 10, 4, 7, 10, 10
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OFFSET

1,1


COMMENTS

Inspired by and analogous to A305382.
Just as for A305382, it is conjectured that a(n) is finite for all n.
All primes less than prime(125000000) have been checked.
First occurrence of k=2,3,4,...: 2, 1, 4, 5, 7, 11, 15, 19, 26, 58, 80, 125, 169, 121, 67, 525, 808, 938, 1799, 1926, 2760, 10658, 4661, 14433, 47463, 22304, 11878, 32103, 101513, 146448, 249616, 266149, 2580007, 2060718, 2883547, 11483667, 8388622, 19786313, ..., .


LINKS

Table of n, a(n) for n=1..100.


EXAMPLE

a(5) = 5 because the 5th prime, 11 > 21 > 3 & 7 > 5 & 13 > 9 & 25 > 3 & 5. Thus there are 5 primes in order of appearance {5, 11, 3, 7, 13}.


MATHEMATICA

g[lst_List] := Union@ Join[lst, First@# & /@ Flatten[ FactorInteger[2lst 1], 1]]; f[n_] := Length@ NestWhile[g@# &, {Prime@n}, UnsameQ, All]; Table[ f[n], {n, 100}]


PROG

(PARI) a(n) = {va = [prime(n)]; done = 0; while (! done, done = 1; for (k=1, #va, f = factor(2*va[k]1); for (j=1, #f~, if (! vecsearch(va, f[j, 1]), va = Set(concat(va, f[j, 1])); done = 0); ); ); ); #select(x>isprime(x), va); } \\ Michel Marcus, Jul 03 2018


CROSSREFS

Cf. A305382.
Sequence in context: A235354 A058743 A117970 * A308194 A245572 A344985
Adjacent sequences: A298901 A298902 A298903 * A298905 A298906 A298907


KEYWORD

nonn


AUTHOR

Robert G. Wilson v, Jun 18 2018


STATUS

approved



