A298908 Smallest composite k such that (n^k - 1)/(n - 1) == 1 (mod k) for n > 1.

341, 91, 4, 15, 6, 25, 4, 9, 10, 33, 4, 65, 14, 15, 4, 9, 6, 49, 4, 21, 22, 69, 4, 25, 9, 9, 4, 15, 6, 49, 4, 33, 34, 9, 4, 133, 38, 15, 4, 21, 6, 25, 4, 9, 46, 65, 4, 25, 10, 39, 4, 9, 6, 35, 4, 25, 58, 15, 4, 91, 9, 9, 4, 15, 6, 49, 4, 15, 10, 9, 4, 65, 15, 15, 4, 21, 6, 49, 4, 9

Offset: 2

Author

Thomas Ordowski and Robert G. Wilson v, Jan 28 2018

Keywords

• nonn

Comments

The smallest repunit pseudoprime to base n.

a(n) is the smallest composite k such that n^k == n (mod (n-1)k).

a(n) is the smallest composite k such that (n^k - 1)/(n - 1) is a Fermat pseudoprime to base n.

a(n) >= A000790(n).

a(n) <= A271801(n).

a(m!+1) > m.

a(4m) = 4.

Records: 341, 361, 403, 561, 685, 1247, 1387, 1891, 2353, 2701, 3277, 4681, 5173, 5461, 6001, 6541, 7445, ..., .

If n is composite, then a(n) <= n. There are only finitely many primes p such that a(p) > p. It seems that a(n) < n for all sufficiently large n. - Thomas Ordowski, Sep 10 2018

Links

• Robert G. Wilson v, Table of n, a(n) for n = 2..10000

Crossrefs

Cf. A000790, A271801.

Programs

Mathematica

f[n_] := Block[{k = 4}, While[PrimeQ@k || Mod[(n^k -1)/(n -1), k] != 1, k++]; k]; Array[f, 80, 2]
With[{r = Select[Range[4, 400], CompositeQ]}, Table[SelectFirst[r, Mod[(n^# - 1)/(n - 1), #] == 1 &], {n, 2, 81}]] (* Michael De Vlieger, Jan 28 2018 *)