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A298904
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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, subtract 1, find all prime factors) with those primes; a(n) = -1 if this produces infinitely many primes.
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0
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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
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1,1
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COMMENTS
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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, ..., .
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LINKS
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EXAMPLE
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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}.
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MATHEMATICA
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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}]
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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