A298904 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.

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

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: A058743 A117970 A364846 * A308194 A245572 A344985

Adjacent sequences: A298901 A298902 A298903 * A298905 A298906 A298907

Keyword

nonn

Author

Robert G. Wilson v, Jun 18 2018

Status

approved