My favorite "1 not behaving as a prime" example.<p>Here are two simple javascript functions:<p>function f(n,k){
var t = 0;
for( var j = 2; j <= n; j++ )t += 1/k - f(n/j, k+1);
return t;}
function p(n){
return f(n,1)-f(n-1,1);
}<p>If you call p(n) when n is prime, it will return 1.
If you call p(n) when n is a prime power (so, say, 4 or 9 or 16), it will return 1/power (so p(4) is .5, p(8) is .3333..., etc).
If you call p(n) with a number with multiple prime bases (so 6 or 14 or 30 or...), it will return 0.<p>And if you call p(1), it will return 0, NOT 1.<p>In fact, f(n,1) here is a compact (and slow) way of computing the Riemann Prime Counting function: <a href="http://mathworld.wolfram.com/RiemannPrimeCountingFunction.html" rel="nofollow">http://mathworld.wolfram.com/RiemannPrimeCountingFunction.ht...</a><p>Another way to compute this exact same function (given as (8) on that link) uses the famous Riemann Zeta function zeroes, although that is much harder to follow.<p>Now, the behavior of the Riemann Prime Counting function doesn't PROVE that 1 isn't a prime, which, as noted, is a question about definition. But what it does do is show that, in an extremely important context, a context that seems to be, mathematically, solely about identifying primes, 1 isn't behaving like the primes at all.