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).

0, 1, 2, 3, 5, 13, 49, 529, 21121, 10369921, 213952189441, 2214253468601687041, 473721461635593679669210030081, 1048939288228833101089604217183056027094304481281

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