Is it a Satisfying Number?

inspired by this chat conversation

A satisfying number is a number whose decimal representation is of the form abx, with the following properties:

• x is the longest trailing repeating suffix, or the last digit if there is no repetition at the end (123333 -> 3333, 545656 -> 5656, 123 -> 3)
• b is the single digit prior to x (123333 -> 2, 55545656 -> 4)
• a is the remaining prefix (123333 -> 1, 55545656 -> 555)
• a == c**b (** denotes exponentation), where c is the number of repetitions of the smallest repeating portion of x (1623333 -> 4 (3 3 3 3, not 33 33))

For example, 8300 is a satisfying number with a = 8, b = 3, c = 2, and x = 00. 24651 is not a satisfying number, because x = 1, b = 5, a = 246, and there is no integer c that satisfies c^5 = 246. 1222 is also not a satisfying number, because with x = 222 and b = 1, there are no remaining digits for a.

Given a positive integer n >= 100, output whether or not n is a satisfying number.

Examples

8300: True (a=8, b=3, c=2, x=00)
24651: False
1222: False
92555: True (a=9, b=2, c=3, x=555)
64633: True (a=64, b=6, c=2, x=33)
512944: True (a=512, b=9, c=2, x=44)
123: True (a=1, b=2, c=1, x=3)
822809: False
376664: False
723799: False
1234: False
34330000000: True (a=343, b=3, c=7, x=0000000)
92313131: True (a=9, b=2, c=3, x=313131)
16424442444: True (a=16, b=4, c=2, x=24442444)
• Sandbox – Mego Dec 22 '17 at 19:00
• Also somewhat related. – Mr. Xcoder Dec 22 '17 at 19:12
• With 8333 is x,c,b,a=33,2,3,8 and therefore satisfying? – Jonathan Allan Dec 22 '17 at 20:18
• @JonathanAllan No, because x is greedy. – Mego Dec 22 '17 at 20:18
• @JonathanAllan Exactly. A number having at least two digits before a repeating portion is a necessary condition for being satisfying. – Mego Dec 22 '17 at 20:21

Jelly, 26 bytes

feels too long

DŒṖṖEÐƤḄ$ÐṀLÐṂṪµḢ*@0¦LµṪ⁼Ḍ A monadic link taking an integer and returning 1 if the input is satisfying and 0 if not. Try it online! or see a test-suite How? DŒṖṖEÐƤḄ$ÐṀLÐṂṪµḢ*@0¦LµṪ⁼Ḍ - Link: integer, n    e.g. 8300
D                          - to decimal list          [8,3,0,0]
ŒṖ                        - all partitions           [[,,,],[,,[0,0]],[,[3,0],],[,[3,0,0]],[[8,3],,],[[8,3],[0,0]],[[8,3,0],],[[8,3,0,0]]]
Ṗ                       - pop (the "no tail" one)  [[,,,],[,,[0,0]],[,[3,0],],[,[3,0,0]],[[8,3],,],[[8,3],[0,0]],[[8,3,0],]]
ÐṀ                - keep maximal under the operation: e.g. for [[8,3],,]

Try it online!

Python 3, 101 bytes

import re
x,_,b,a=re.findall(r"((.+?)\2+)(.)(.*)",input()[::-1])
if int(a[::-1])!=len(x)**int(b):d

Python 3, 107 bytes

import re
x,_,b,a=re.findall(r"((.+?)\2+)(.)(.*)",input()[::-1])
if int(a[::-1])!=len(x)**int(b):int('')

Output is by exit code.

This code doesn't run properly on Tio due to a range error. Works perfectly in IDLE.

JavaScript (ES6), 282 268 bytes

t=(d,n,l,i,r)=>d.slice((m=-l*i)-l,m).join!=n?r:t(d,n,l,i+1,{p:r.p+n,r:r.r+1});p=(d,l,r)=>l<1?r:p(d,l-1,r.r<(z=t(d,(m=d.slice(-l).join),l,1,{p:m,r:1})).r&&(z.r>1|l==1)?z:r);u=n=>(d=[...n]).slice(0,(q=(s=d.length)-(m=p(d,s,{p:"",r:0})).p.length-1)).join==m.r**d[q]

function onChange() {
var value = document.getElementById("input").value;
console.log("%s => %s", value, u(value));
}
<input id="input" type="number" onchange="onChange()" />

Python 2, 286 bytes

yeesh.

n=input()
N=lambda p,l=0:N(n[:-len(p)],p,l+1)if n[-len(p):]==p else l
try:
s=[(N(n[-i-1:]),n[-i-1:])for i,_ in enumerate(n)if N(n[-i-1:])!=1]
if len(s)==0:s=[(1,n[-1])]
q=max(s,key=lambda a:len(a*a))
i=len(q*q)
print n[:-i-1]==min(s)**int(n[-i-1])
except:print 0

N is a recursive function that finds the number of times a suffix substring is repeated in a string. This basically loops through all possible suffixes, finding the number of times each is repeated using N; this excludes all values where N==1 because they refer to no repeats; if the list ends up being empty, the suffix of the last character is appended to the list.

Then, the longest suffix is taken, (q), the number of characters it takes up is found (i), and a==c**b is checked (print ...).

If an error occurs along the way (which it often will), it is caught in the except block.

Any suggestions are more than welcome!