|
|
A067849
|
|
a(n) = max{k: f(n),...,f^k(n) are prime}, where f(m) = 2m+1 and f^k denotes composition of f with itself k times.
|
|
3
|
|
|
2, 4, 1, 0, 3, 1, 0, 1, 1, 0, 2, 0, 0, 2, 1, 0, 0, 1, 0, 3, 1, 0, 1, 0, 0, 2, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 2, 0, 0, 6, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 2, 0, 0, 5, 1
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
If a(n) > 3 and n > 5, then the final digit of n is 4 or 9.
a(n) > 0 if and only if n appears in A005097.
More generally, a(n) > m if and only if all of 2^k(n+1) - 1 for 0 <= k <= m are in A005097.
Creating a tile labeled by a multiple of p for a prime p with a relatively large value of a(p) is considered valuable in the game DIVE (see links). (End)
|
|
LINKS
|
Number-theoretic puzzle game DIVE
|
|
EXAMPLE
|
f(2) = 5, f(f(2)) = 11, f(f(f(2))) = 23, f(f(f(f(2)))) = 47, all prime, but f^5(2) = 95 is not prime, so a(2) = 4.
|
|
MATHEMATICA
|
f[n_] := Module[{a = 2n + 1, i = 0}, While[PrimeQ[a], i++; a = 2a + 1]; i]; Table[f[i], {i, 1, 60}]
|
|
PROG
|
(PARI) a(n) = {my(nb = 0, newn); while (isprime(newn=2*n+1), nb++; n = newn); nb; } \\ Michel Marcus, Nov 10 2018
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|