A000236

[Max]

Max wrote:
>> The author defines another function $\Lambda^*(k,m)$ obtained by 
>> deleting the condition of divisibility in (1)
>> so that one asks only for the first appearance of $m$ consecutive 
>> integers whose $k$th power characters are identical.
> 
> As I understood, these m consecutive integers are all either k-powers or 
> k-nonpowers modulo p.

I was wrong here. k-th power character is not just "yes/no" value.
I believe it is defined as
J(a,k,p) = a^((p-1)/gcd(p-1,k)) mod p

>> $\Lambda^*(k,2)$ turns out to be very much smaller. The author has 
>> established that $\Lambda^*(2,2)=3$, $\Lambda^*(3,2)=8$, 
>> $\Lambda^*(4,2)=20$,
>> $\Lambda^*(5,2)=44$, $\Lambda^*(6,2)=80$, $\Lambda^*(7,2)=343$. The 
>> proofs are short enough to be done "by hand", although the last two 
>> proofs are omitted to save space. 
> 
> This is A000236 and its definition again is very cryptic.
> I suggest to replace it with something like ``A000236(n)=minimum t such 
> that for every prime p there exists a pair (r,r+1) of consecutive 
> n-powers or n-nonpowers modulo p with r<=t''.

Should be ``minimum t such that for every prime p there exists a pair r,r+1 having the same n-th power characters modulo p with r<=t''.

> And again there exists a satellite sequence a(n)=minimum prime p for 
> which the smallest pair of consecutive n-powers or n-nonpowers modulo p 
> starts with A000236(n).

Respectively, a(n)=minimum prime p for which the smallest pair of consecutive residues having the same n-th power characters
modulo p starts with A000236(n).

> Could anybody please compute this sequence as well? It starts with 11.

I've found the first three terms:  11, 67, 24077
This sequence is not in OEIS.

Max