The phrase "nonconsecutive integer" does not make sense. Only two or more integers can be consecutive or not.
"If a+b is an odd number then a and b are not consecutive odd integers" is correct.
"If a and b are consecutive odd numbers, then a+b is an even number." I was thinking it was "If a+b is an odd number then a or b is a nonconsecutive even integer". I thought this mean the same as "If a+b is an odd number then a and b are not consecutive odd integers"? I was thinking these two sentences mean the same thing because either a or b or both are nonconsective even integers?
No, for example, the following pairs of numbers are not consecutive odd integers: (1, 5), (1, 2), (2, 4) and (2, 6). Of those pairs, only the last one consists of even integers that are not consecutive.A pair of numbers does not consist of consecutive odd integers either when the numbers are odd but not consecutive, or if at least one of them is even.