A280926 Least k such that the first n digits of the decimal expansion of the ratio of the perimeter of a regular k-gon to its diameter match those of Pi.

5, 7, 29, 47, 119, 699, 1407, 4911, 18971, 46803, 119951, 363209, 1276197, 3722389, 19973297, 73605289, 183273481, 390720475, 1671075265, 4541314567, 22107473795, 44810965685, 172567099183, 617945607281, 1835952288687, 3938674815741, 19847928172101

Offset: 1

Author

Rick L. Shepherd, Jan 10 2017

Keywords

• nonn
• base
• more

Comments

By definition, the diameter of a regular k-gon is the length of its longest diagonal.

All terms are odd; see Formula section. - Jon E. Schoenfield, Mar 29 2021

Examples

			An equilateral triangle (k=3) has no diagonals, and a square (k=4) has perimeter/diameter = sqrt(8) = 2.828427..., but a regular pentagon (k=5) has perimeter/diameter = (5/2)*(sqrt(5) - 1) = 3.090169..., whose first digit (3) matches that of Pi = 3.141592..., so a(1)=5. - _Jon E. Schoenfield_, Mar 31 2021
This ratio for a regular 7-gon (heptagon) is 3.115293... (A280533), where 3.1 equals the first two digits of Pi's decimal expansion. Because the first two digits are not 3.1 for k < 7, a(2) = 7.
		

Crossrefs

Cf. A000796, A280533.

Formula

a(n) = 1 + 2*floor((1/2)*(1 + sqrt((Pi^3/24)/(Pi-floor(Pi*10^(n-1))/10^(n-1))))). - Jon E. Schoenfield, Mar 28 2021

Extensions

a(13)-a(27) from Jon E. Schoenfield, Mar 28 2021