My 2 cents worth of suggestions...
This line:
digitalRead(button);
doesn't do anything. It's probably a leftover. It should be deleted.
Use more meaningful names for variables. For example:
const int btnPin = 7;
const int ledPin1 = 9;
const int ledPin2 = 10;
const int ledPin3 = 11;
const int ledPin4 = 13;
int btnState = 0;
int numBlinkCycles = 0;
int analogPin = 0;
Also, comments should not be a description of what the code line does, but rather why it does it or how it relates to the problem domain.
Use the handy map() function:
numBlinkCycles = analogRead(analogPin)
numBlinkCycles = map(numBlinkCycles, 0, 1023, 0, 60);
Use named constants:
const int MIN_DELAY_CNT = 5;
if (numBlinkCycles < MIN_DELAY_CNT) {
numBlinkCycles = MIN_DELAY_CNT;
}
When one uses "good" variable and named constant names, comments often become superfluous. And that means the code is more readable. Which is a Good Thing (TM).
Looks reasonable to me. The delays will probably debounce it for you.
When value1 is 60, the sketch samples the button once in 2 minutes. IOW, the button pin has to read HIGH when the blink sequence ends and loop() restarts, or it won't be detected. This is sub optimal IMHO, however it could be OK in this particular application (it depends on the hardware context).
HTH