A222298 -id:A222298 - cp's OEIS frontend

A222300 Length of the closed curve through Gaussian primes described in A222298.

32, 48, 1316, 72, 1536, 168, 168, 152, 152, 1536, 140, 352, 352, 132, 172, 280, 648, 132, 92, 12, 96, 32, 332, 332, 460, 30492, 652, 328, 460, 30492, 748, 236, 64, 112, 204, 336, 336, 24560, 24560, 448, 440, 13016, 1536, 316, 108, 2224, 132, 116, 864, 80, 1128

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 A222298 for the number of line segments between primes.

			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}. This loop is 168 units long.

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

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]; Total[Abs[Differences[lst]]], {p, cp}]

A222299 Number of different Gaussian primes in the Gaussian prime spiral beginning at the n-th positive real Gaussian prime (A002145).

Original entry on oeis.org

8, 10, 172, 12, 168, 19, 19, 21, 21, 168, 14, 37, 37, 14, 18, 30, 68, 10, 10, 4, 10, 4, 29, 29, 32, 2484, 58, 30, 32, 2484, 76, 16, 10, 10, 18, 23, 23, 1861, 1861, 30, 34, 958, 126, 22, 10, 182, 10, 10, 74, 10, 112, 26, 48, 29, 29, 774, 13, 13, 26, 774, 18, 10

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 A222298 for the number of line segments between primes.

			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}. But only 19 are unique.

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

loop2[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[Union[lst]]]; cp = Select[Range[1000], PrimeQ[#, GaussianIntegers -> True] &]; Table[loop2[p], {p, cp}]

A222594 Length of the Gaussian prime spiral beginning at the n-th first-quadrant Gaussian prime (A222593).

Original entry on oeis.org

4, 28, 28, 4, 12, 28, 28, 12, 4, 12, 4, 28, 12, 4, 12, 100, 4, 100, 12, 12, 28, 28, 12, 28, 28, 4, 260, 12, 12, 100, 12, 12, 100, 100, 4, 12, 4, 12, 260, 4, 4, 12, 260, 100, 12, 260, 260, 4, 4, 260, 260, 260, 100, 12, 100, 28, 260, 4, 12, 100, 12, 12, 260

Offset: 1

T. D. Noe, Feb 27 2013

nonn

This is the idea of A222298 extended to first-quadrant Gaussian primes (A222593). It appears that all multiples of 4 eventually appear as a length.

			The smallest such prime is 1 + i. The spiral is {1 + i, 2 + i, 2 - i, 1 - i, 1 + i}, which consists of only Gaussian primes.

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..2829

Cf. A222298 (spiral lengths beginning at the n-th positive real Gaussian prime).

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]]; nn = 20; ps = {}; Do[If[PrimeQ[i + (j - i) I, GaussianIntegers -> True], AppendTo[ps, i + (j-i)*I]], {j, 0, nn}, {i, 0, j}]; Table[loop[ps[[n]]] - 1, {n, Length[ps]}]

A222595 Number of different Gaussian primes in the Gaussian prime spiral beginning at the n-th first-quadrant Gaussian prime (A222593).

Original entry on oeis.org

4, 24, 24, 4, 8, 22, 22, 8, 4, 8, 4, 22, 8, 4, 10, 92, 4, 92, 10, 10, 22, 22, 10, 22, 22, 4, 172, 10, 10, 92, 10, 10, 92, 92, 4, 10, 4, 10, 172, 4, 4, 10, 172, 92, 10, 172, 172, 4, 4, 172, 172, 172, 92, 10, 92, 28, 172, 4, 12, 92, 10, 10, 172, 92, 4, 12, 172, 28

Offset: 1

T. D. Noe, Feb 27 2013

nonn

This is the idea of A222299 extended to first-quadrant Gaussian primes. The first odd number is a(79) = 29.

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..2829

Cf. A222298 (spiral lengths beginning at the n-th positive real Gaussian prime).

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]]; nn = 20; ps = {}; Do[If[PrimeQ[i + (j - i) I, GaussianIntegers -> True], AppendTo[ps, i + (j-i)*I]], {j, 0, nn}, {i, 0, j}]; Table[loop[ps[[n]]]; Length[Union[lst]], {n, Length[ps]}]

A222596 Length of the closed curve through Gaussian primes described in A222594.

Original entry on oeis.org

6, 64, 64, 8, 32, 92, 92, 32, 8, 32, 12, 92, 32, 12, 48, 412, 12, 412, 48, 48, 92, 92, 44, 92, 92, 12, 1316, 48, 44, 412, 48, 48, 412, 412, 24, 44, 24, 48, 1316, 12, 8, 48, 1316, 412, 44, 1316, 1316, 12, 12, 1316, 1316, 1316, 412, 44, 412, 204, 1316, 28, 72, 412

Offset: 1

T. D. Noe, Feb 27 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 A222594 and A222595 for the number of line segments between primes.

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..2829

Cf. A222298 (spiral lengths beginning at the n-th positive real Gaussian prime).

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]]; nn = 20; ps = {}; Do[If[PrimeQ[i + (j - i) I, GaussianIntegers -> True], AppendTo[ps, i + (j-i)*I]], {j, 0, nn}, {i, 0, j}]; Table[loop[ps[[n]]]; Total[Abs[Differences[lst]]], {n, Length[ps]}]

cp's OEIS Frontend

A222300 Length of the closed curve through Gaussian primes described in A222298.

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A222299 Number of different Gaussian primes in the Gaussian prime spiral beginning at the n-th positive real Gaussian prime (A002145).

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A222594 Length of the Gaussian prime spiral beginning at the n-th first-quadrant Gaussian prime (A222593).

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A222595 Number of different Gaussian primes in the Gaussian prime spiral beginning at the n-th first-quadrant Gaussian prime (A222593).

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A222596 Length of the closed curve through Gaussian primes described in A222594.

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