A078784 Primes on axis of Ulam square spiral (with rows ... / 7 8 9 / 6 1 2 / 5 4 3 / ... ) with origin at (1).

2, 11, 19, 23, 53, 61, 127, 139, 151, 163, 233, 281, 431, 541, 613, 743, 827, 977, 1009, 1279, 1621, 1871, 2003, 2281, 2377, 2731, 3109, 3221, 3511, 3571, 3631, 3691, 4001, 4129, 4523, 4591, 5077, 6361, 6521, 7789, 7877, 8419, 9851, 10151, 10973, 11503, 11719, 11827, 12377, 12601, 12713, 13399

Offset: 1

Author

Donald S. McDonald, Jan 10 2003

Keywords

• nonn

Comments

Quadrants are numbered clockwise: 4=north, 1=east, 2=south, 3=west. The spiral numbers falling on axes (whether prime or not) are 4=north (2n+1)^2-n, 1=east (2n+1)^2+n+1, 2=south (2n)^2-(n-1), 3=west (2n)^2+n+1.

Primes to the left, right, above or below the 1 in the example in A054552.

This is the union of the primes in A168022, A168023, A168025 and A168027. - R. J. Mathar, Jul 11 2014

Examples

			For n=0, quadrant = 1, a(1) =  2, distance = 1;
for n=1, quadrant = 1, a(2) = 11, distance = 2;
for n=2, quadrant = 3, a(3) = 19, distance = 2.
		

Crossrefs

Cf. A039823, A054552, A054556, A054567, A033951, A172979.

Programs

Mathematica

Select[ Sort@ Flatten@ Table[ 4n^2 + (2j - 3)n + 1, {j, 0, 3}, {n, 58}], PrimeQ] (* Robert G. Wilson v, Jul 10 2014 *)

Formula

Primes in A039823(n) = ceiling((n^2 + n + 2)/4). - Georg Fischer, Dec 04 2024

Extensions

a(12) onward from Robert G. Wilson v, Jul 10 2014