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 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':. 0
 47, 227, 71, 359, 113, 563, 173, 839, 251, 1187, 347, 1607, 461, 2099, 593, 2663, 743, 3299, 911, 4007 (list; graph; refs; listen; history; text; internal format)
 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 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

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Last modified April 3 22:48 EDT 2020. Contains 333207 sequences. (Running on oeis4.)