A332045 Numbers k such that ceiling(Pi/arctan(1/k)) = ceiling(k*Pi)+1.

6, 7, 14, 21, 28, 113, 226, 339, 452, 565, 678, 791, 904, 1017, 1130, 1243, 1356, 1469, 1582, 1695, 1808, 1921, 33215, 99532, 364913, 729826, 1725033, 3450066, 5175099, 27235615, 52746197, 131002976, 471265707, 811528438, 2774848045, 4738167652, 567663097408

Offset: 1

Author

Jianing Song, Feb 05 2020

Keywords

• nonn

Comments

Note that ceiling(Pi/arctan(1/k)) - ceiling(k*Pi) is equal to either 0 or 1, that is, for all other k we have ceiling(Pi/arctan(1/k)) = ceiling(k*Pi).

Numbers k such that there exists some integer m such that Pi/arctan(1/k) > m > k*Pi.

Numbers k such that A331859(k^2) = A121854(k^2)+1 = A121855(k^2).

In A331859 there is a remark that A331859(100^n) = A011545(n). I'm in doubt of this, because if k = 10^n is here, then A331859(100^n) = ceiling(k*Pi), while A011545(n) = ceiling(k*Pi)-1, this equality would be violated.

Note that for k >= 3 we have 1/k < Pi/arctan(1/k)-k*Pi < (Pi/3)/k. As a result, a necessary condition for k being a term here is that there exists some m such that 0 < m/k - Pi < (Pi/3)/k^2, and a sufficient condition is that there exists some m such that 0 < m/k - Pi < 1/k^2.

Let P(n) = A002485(n), Q(n) = A002486(n), then it is known that 1/(Q(n)*(Q(n)*Q(n+1))) < |P(n)/Q(n) - Pi| < 1/(Q(n)*Q(n+1)) for n >= 2; furthermore, P(n)/Q(n) - Pi is positive for odd n and negative for even n. As a result, let n >= 3, then we have:

- If n is even, then Q(n) can never be a term.

- If n is odd, then k = Q(n)*t is a term if t <= sqrt(Q(n+1)/Q(n)), in which case ceiling(Pi/arctan(1/k)) = P(n)*t+1 and ceiling(k*Pi) = P(n)*t. The converse is not true (e.g., n = 3, t = 4745).

Examples

			Pi/arctan(1/6) = 19.0228..., 6*Pi = 18.8495..., so 6 is a term.
113*t is here for t <= 17, because ceiling(Pi/arctan(1/(113*t))) = 355*t+1 and ceiling((113*t)*Pi) = 355*t.
		

Links

• Jon E. Schoenfield, Table of n, a(n) for n = 1..2259
• Jon E. Schoenfield, Magma program
• Wikipedia, Continued fraction
• Zhihu, A counting puzzle of colliding blocks which is related to Pi (in Chinese)

Crossrefs

Cf. A121381 (ceiling(n*Pi)), A121854 (floor(sqrt(n)*Pi)), A121855 (ceiling(sqrt(n)*Pi)), A011545 (floor(10^n*Pi)).

Cf. A331859, A002486, A002485.

Programs

Magma

// See Schoenfield link.

PARI

default(realprecision, 10000); isA332045(n) = ceil(Pi/atan(1/n))!=ceil(n*Pi)

Extensions

a(27)-a(32) from Jon E. Schoenfield, Feb 12 2020

a(33)-a(36) from Giovanni Resta, Feb 12 2020

a(37) from Jon E. Schoenfield, Feb 15 2020