A141537 - OEIS

%I #6 Oct 02 2016 09:54:56

%S 47,227,71,359,113,563,173,839,251,1187,347,1607,461,2099,593,2663,

%T 743,3299,911,4007

%N An example of a simple prime-generating formula similar to Rowland's ( a(1) = 7, n>1, a(n) = a(n-1) + gcd(n,a(n-1) ) that is a particular instance of a more general formula. The sequence submitted is the first 20 values that do not equal '1':.

%C Below is a general expression that can be used as a starting point for finding similar formulas and three examples.

%C 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.

%C 1) initialize the integers x,k,a,b and choose f(x), f(k).

%C 2) repeat indefinitely a two-step process:

%C x := x + 1,

%C If GCD( f(x), f(x - 1) - a* f(k) ) > 1, then k := k + b;

%C Examples:

%C A) f(x) := 5*x^2 + 5*x + 1, f(k) = k, x = 1, k = 2, a = 10, b = 1;

%C 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

%C B) f(x) := x^2 - x + 41, f(k) := k, x = 1, k = 2, a = 3, b = 1;

%C 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 C) f(x) := 5*x^2 + 5*x + 1, f(k) = x^2 - x + 41, x = 1, k = 2, a = 2, b = 1;

%C 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

%D Eric S. Rowland, A simple prime-generating recurrence, Abstracts Amer. Math. Soc., 29 (No. 1, 2008), p. 50 (Abstract 1035-11-986).

%F 1) initialize the integers x,k,a,b and choose f(x), f(k). 2) repeat indefinitely a two-step process: x := x + 1, If GCD( f(x), f(x - 1) - a* f(k) ) > 1, then k := k + b; In the above formual let f(x) := x^2 - x + 41, f(k) := k, x = 1, k = 2, a = 3, b = 1.

%Y Cf. A084662, A137613.

%K nonn,uned

%O 1,1

%A Aldrich Stevens (aldrichstevens(AT)msn.com), Aug 15 2008