A222300 -id:A222300 - cp's OEIS frontend

A270641 The sequence a of 1's and 2's starting with (1,1,1,1) such that a(n) is the length of the (n+1)st run of a.

1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2

Offset: 1

Clark Kimberling, Apr 05 2016

Guide to related sequences (with adjustments for initial terms):
1, 1, 1, 1;    a(n) = length of (n + 1)st run of a; A270641
1, 1, 1, 2;    a(n) = length of (n + 2)nd run of a; A270641
1, 1, 2, 1;    a(n) = length of (n + 3)rd run of a; A270641
1, 1, 2, 2;    a(n) = length of (n + 2)nd run of a; A270642
1, 2, 1, 1;    a(n) = length of (n + 3)rd run of a; A022300
1, 2, 1, 2;    a(n) = length of (n + 4)th run of a; A270641
1, 2, 2, 1;    a(n) = length of (n + 3)rd run of a; A270643
1, 2, 2, 2;    a(n) = length of (n + 2)nd run of a; A270644
2, 1, 1, 1;    a(n) = length of (n + 2)nd run of a; A270645
2, 1, 1, 2;    a(n) = length of (n + 3)rd run of a; A222300
2, 1, 2, 1;    a(n) = length of (n + 4)th run of a; A270641
2, 1, 2, 2;    a(n) = length of (n + 3)rd run of a; A000002 (Kolakoski)
2, 2, 1, 1;    a(n) = length of (n + 2)nd run of a; A270646
2, 2, 1, 2;    a(n) = length of (n + 3)rd run of a; A270647
2, 2, 2, 1;    a(n) = length of (n + 2)nd run of a; A270644
2, 2, 2, 2;    a(n) = length of (n + 1)st run of a; A270648

			a(1) = 1, so the 2nd run has length 1, so a(5) must be 2 and a(6) = 1.
a(2) = 1, so the 3rd run has length 1, so a(7) = 2.
a(3) = 1, so the 4th run has length 1, so a(8) = 1.
a(4) = 1, so the 5th run has length 1, so a(9) = 2.
a(5) = 2, so the 6th run has length 2, so a(10) = 2 and a(11) = 1.
Globally, the runlength sequence of a is 4,1,1,1,1,2,1,2,1,2,2,1,...., and deleting the first term leaves a = A270641.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A000002, A006928, A022300,

a = {1, 1, 1, 1};
Do[a = Join[a, ConstantArray[If[Last[a] == 1, 2, 1], {a[[n]]}]], {n,   200}]; a
(* Peter J. C. Moses, Apr 01 2016 *)

A222298 Length of the Gaussian prime spiral beginning at the n-th positive real Gaussian prime (A002145).

Original entry on oeis.org

12, 12, 260, 12, 236, 28, 28, 28, 28, 236, 20, 44, 44, 20, 20, 36, 76, 12, 12, 4, 12, 4, 36, 36, 36, 3276, 76, 36, 36, 3276, 84, 20, 12, 12, 20, 36, 36, 2444, 2444, 36, 44, 1356, 156, 28, 12, 220, 12, 12, 84, 12, 132, 28, 68, 36, 36, 1044, 20, 20, 28, 1044, 20

Offset: 1

T. D. Noe, Feb 25 2013

nonn

The Gaussian prime spiral is described in the short note by O'Rourke and Wagon. It is not known if every iteration is a closed loop. See A222299 for the number of distinct primes on the spiral. See A222300 for the length of the spiral (which is the same as the number of numbers tested for primality, without memory).

This idea can be extended to any Gaussian prime. Sequences A222594, A222595, and A222596 show the results for first-quadrant Gaussian primes. - T. D. Noe, Feb 27 2013

			The loop beginning with 31 is {31, 43, 43 - 8i, 37 - 8i, 37 - 2i, 45 - 2i, 45 - 8i, 43 - 8i, 43, 47, 47 - 2i, 45 - 2i, 45 + 2i, 47 + 2i, 47, 43, 43 + 8i, 45 + 8i, 45 + 2i, 37 + 2i, 37 + 8i, 43 + 8i, 43, 31, 31 + 4i, 41 + 4i, 41 - 4i, 31 - 4i, 31}. The first and last numbers are the same. So only one is counted.

Joseph O'Rourke and Stan Wagon, Gaussian prime spirals, Mathematics Magazine, vol. 86, no. 1 (2013), p. 14.

T. D. Noe, Table of n, a(n) for n = 1..1000
T. D. Noe, Plot beginning with 11 (similar to the cover of Mathematics Magazine, vol. 86, no. 1 (2013))
Joseph O'Rourke, MathOverflow: Gaussian prime spirals

loop[n_] := Module[{p = n, direction = 1}, lst = {n}; While[While[p = p + direction; ! PrimeQ[p, GaussianIntegers -> True]]; direction = direction*(-I); AppendTo[lst, p]; ! (p == n && direction == 1)]; Length[lst]]; cp = Select[Range[1000], PrimeQ[#, GaussianIntegers -> True] &]; Table[loop[p]-1, {p, cp}]

cp's OEIS Frontend

A270641 The sequence a of 1's and 2's starting with (1,1,1,1) such that a(n) is the length of the (n+1)st run of a.

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