
COMMENTS

Below is a general expression that can be used as a starting point for finding similar formulas and three examples.
Be forewarned that not every possibility will work  additional conditions may apply but it is easy to see that there are no doubt actually an infinite number of algorisms much like Rowland's that will have hundreds or thousands of primes between the 1's before a composite is encountered.
1) initialize the integers x,k,a,b and choose f(x), f(k).
2) repeat indefinitely a twostep process:
x := x + 1,
If GCD( f(x), f(x  1)  a* f(k) ) > 1, then k := k + b;
Examples:
A) f(x) := 5*x^2 + 5*x + 1, f(k) = k, x = 1, k = 2, a = 10, b = 1;
the first 20 values of the sequence that do not equal one: 11, 31, 61, 101, 151, 211, 281, 19, 41, 29, 661, 11, 911,1051,1201,1361,1531,59,1901
B) f(x) := x^2  x + 41, f(k) := k, x = 1, k = 2, a = 3, b = 1;
the first 20 values of the sequence that do not equal one: 47, 227, 71, 359, 113, 563, 173, 839, 251,1187,347, 1607, 461,2099,593, 2663, 743,3299, 911,4007
C) f(x) := 5*x^2 + 5*x + 1, f(k) = x^2  x + 41, x = 1, k = 2, a = 2, b = 1;
the first 20 values of the sequence that do not equal one: 11, 1979, 2549, 11,4691, 11, 8929, 29, 11, 22051, 41, 19, 48619, 61751, 11, 229, 11, 144779, 175141, 11
