A141537 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':.

47, 227, 71, 359, 113, 563, 173, 839, 251, 1187, 347, 1607, 461, 2099, 593, 2663, 743, 3299, 911, 4007

Offset

1,1

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 two-step 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

References

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

Links

Table of n, a(n) for n=1..20.

Formula

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.

Crossrefs

Cf. A084662, A137613.

Sequence in context: A211335 A142203 A067986 * A102090 A033226 A142946

Adjacent sequences: A141534 A141535 A141536 * A141538 A141539 A141540

Keyword

nonn,uned

Author

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

Status

approved