A124183 a(0) = 0; a(1) = 1; for n >= 2, a(n) = the n-th integer from among those positive integers which are coprime to a(n-1)+a(n-2).

0, 1, 2, 4, 11, 8, 6, 15, 13, 19, 19, 23, 41, 25, 43, 31, 31, 35, 59, 37, 59, 61, 79, 67, 47, 77, 53, 71, 57, 57, 91, 63, 81, 97, 67, 71, 109, 137, 115, 137, 139, 127, 103, 111, 87, 149, 93, 103, 111, 97, 107, 161, 105, 129, 175, 115, 143, 173, 117, 151, 121, 129, 153

Offset

0,3

Comments

For n>=9, a(n-1)+a(n-2) is even so a(n) is odd and >= 2n-1. It appears that a(n)=2n-1 infinitely often, when a(n-1)+a(n-2)=2p with p prime. A good upper bound is not known; presumably a(n)/n is unbounded, since it's large whenever a(n-1)+a(n-2) is divisible by many small primes. - Dean Hickerson, Dec 07 2006

Zak Seidov observed that straight lines are visible in the graph of a(n). The lowest line is given by the points with a(n)=2n-1. There are also many points a short distance above this line; these occur when a(n-1)+a(n-2) is a small power of 2 times a prime. Other lines show points with a(n-1)+a(n-2) equal to a small number times a large prime; e.g. when n>18 and a(n-1)+a(n-2)=6p, a(n) is either 3n+1 or 3n+2. - Dean Hickerson, Dec 07 2006

Links

Zak Seidov, Table of n, a(n) for n = 0..999.

Example

a(6)+a(7)=21. 1,2,4,5,8,10,11,13,16,17,... are the positive integers which are coprime to 21. 13 is the 8th of these integers, so a(8) = 13.

Mathematica

f[l_List] := Block[{k = 0, c = Length[l]}, While[c > 0, k++; While[GCD[k, l[[ -1]] + l[[ -2]]] > 1, k++ ]; c--; ]; Append[l, k]]; Nest[f, {0, 1}, 65] (* Ray Chandler, Dec 06 2006 *)
a[0]=0; a[1]=1; a[n_]:=a[n]=Module[{s, c, v}, s=a[n-1]+a[n-2]; v=0; For[c=1, c<=n, c++, While[GCD[ ++v, s]>1, Null]]; v] (* _Hickerson_* )

Crossrefs

Cf. A121047.

Sequence in context: A356175 A133757 A246164 * A218643 A300375 A075488

Adjacent sequences: A124180 A124181 A124182 * A124184 A124185 A124186

Keyword

nonn

Author

Leroy Quet, Dec 05 2006

Extensions

Extended by Ray Chandler and Zak Seidov, Dec 06 2006

Status

approved