A328914 Smallest index m such that from the m-th term on, the sequence {k^k mod A192135(n): k >= 0} enters into a cycle.

3, 3, 3, 3, 4, 7, 4, 7, 7, 4, 7, 11, 7, 11, 7, 11, 6, 11, 7, 15, 7, 15, 6, 15, 7, 15, 6, 19, 7, 19, 13, 6, 19, 19, 13, 8, 23, 6, 13, 23, 23, 8, 16, 11, 23, 16, 27, 11, 27, 8, 16, 27, 27, 16, 11, 8, 31, 16, 31, 11, 31, 16, 8, 31, 11, 22, 35, 35, 22, 8, 35, 16, 35, 22, 39, 8, 16, 25, 39, 39, 25, 12, 16, 39, 15, 25, 43

Offset: 1

Author

Jianing Song, Oct 31 2019

Keywords

• nonn

Comments

Let f(n) be the smallest index m such that from the m-th term on, the sequence {k^k mod n: k >= 0} enters into a cycle, then:

(a) if gcd(n1,n2) = 1, then f(n1*n2) = max{f(n1), f(n2)}. For example, f(648) = max{f(8), f(81)} = 4;

(b) f(1) = 0; for primes p, f(p^e) = 1 if e <= p; f(p^e) is the largest s that is no greater than e and that s-1 is divisible by p but not by p^2.

Proof: If we define b(k) = k^k mod p^e if gcd(k,p) = 1, 0 otherwise, it is easy to see that {b(k): k >= 0} is purely periodic. As a result, for every k >= f(p^e), we have either gcd(k,p) = 1 or p^e | k^k. In other words, let t be the largest number divisible by p such that p^e does not divide t^t, then f(p^e) = t+1.

If p^2 divides t, write t = r*p^2, then v(t^t,p) < v((t+p)^(t+p),p), where t(,p) is the p-adic valuation. This gives r = 0. As a result, f(p^e) is either 1 or a number s such that s-1 is divisible by p but not by p^2. In the latter case, v((s-1)^(s-1),p) = s-1, so s is the largest such number that is no greater than e.

The records in this sequence are {3, 4, 7, 11, 15, 19, 23, ...}

Examples

			A table for f(p^e):
              p
   e  2  3  5  7 11 13
   1  1  1  1  1  1  1
   2  1  1  1  1  1  1
   3  3  1  1  1  1  1
   4  3  4  1  1  1  1
   5  3  4  1  1  1  1
   6  3  4  6  1  1  1
   7  7  7  6  1  1  1
   8  7  7  6  8  1  1
   9  7  7  6  8  1  1
  10  7  7  6  8  1  1
  11 11  7 11  8  1  1
  12 11  7 11  8 12  1
  13 11 13 11  8 12  1
  14 11 13 11  8 12 14
  15 11 13 11 15 12 14
  16 11 16 16 15 12 14
		

Crossrefs

Cf. A192135, A328920 (smallest N such that f(N) = n).

Programs

PARI

b(p,e) = if(!e, 0, if(e<=p, 1, forstep(k=e, p+1, -1, if(k%p==1&&k%(p^2)!=1, return(k)))))
L=List(); my(lim=12); forprime(p=2, lim, for(n=p+1, lim*log(lim)\log(p), listput(L, p^n))); listsort(L); L \\ generates all terms of A192135 below lim^lim
for(k=1, #L, my(p=factor(L[k])[1,1],e=factor(L[k])[1,2]); print1(b(p,e), ", "))

Formula

a(n) = f(A192135(n)), where f is defined in the comment section.