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 A064526 Define a pair of sequences by p(0) = 0, q(0) = p(1) = q(1) = 1, q(n+1) = p(n)*q(n-1), p(n+1) = q(n+1) + q(n) for n > 0; then a(n) = p(n) and A064183(n) = q(n). 13
 0, 1, 2, 3, 5, 13, 49, 529, 21121, 10369921, 213952189441, 2214253468601687041, 473721461635593679669210030081, 1048939288228833101089604217183056027094304481281 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Every nonzero term is relatively prime to all others (which proves that there are infinitely many primes). See A236394 for the primes that appear. LINKS Table of n, a(n) for n=0..13. R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012. - N. J. A. Sloane, Jun 13 2012 Michael Somos and R. Haas, A linked pair of sequences implies the primes are infinite, Amer. Math. Monthly, 110(6) (2003), 539-540. FORMULA a(n) = (a(n-1)^2 + a(n-2)^2 - a(n-1) * a(n-2) * (1 + a(n-2))) / (1 - a(n-2)) for n >= 2. a(n) ~ c^(phi^n), where c = 1.2364241784241086061606568429916822975882631646194967549068405592472125928485... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, May 21 2015 MATHEMATICA Flatten[{0, 1, RecurrenceTable[{a[n]==(a[n-1]^2 + a[n-2]^2 - a[n-1]*a[n-2] * (1+a[n-2]))/(1-a[n-2]), a[2]==2, a[3]==3}, a, {n, 2, 15}]}] (* Vaclav Kotesovec, May 21 2015 *) PROG (PARI) {a(n) = local(v); if( n<3, max(0, n), v = [1, 1]; for( k=3, n, v = [v[2], v[1] * (v[1] + v[2])]); v[1] + v[2])} (PARI) {a(n) = if( n<4, max(0, n), (a(n-1)^2 + a(n-2)^2 - a(n-1) * a(n-2) * (1 + a(n-2))) / (1 - a(n-2)))} CROSSREFS Cf. A001685, A003686, A064183, A064847, A070231, A070233, A070234, A094303. See A236394 for the primes that are produced. Sequence in context: A215102 A110364 A111288 * A261192 A103594 A042695 Adjacent sequences: A064523 A064524 A064525 * A064527 A064528 A064529 KEYWORD nonn,easy AUTHOR Michael Somos, Oct 07 2001 STATUS approved

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Last modified June 12 19:52 EDT 2024. Contains 373360 sequences. (Running on oeis4.)