A022931 Number of e^m between Pi^n and Pi^(n+1).

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

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• easy

Examples

			Pi^5 = 306.01968478528145326274131... and Pi^6 = 961.389193575304437...; in between them we find e^6 = 403.4287934927351226... and no other powers of e with integer exponents. Hence a(5) = 1.
Pi^6 = 961.389193575304437... and Pi^7 = 3020.2932277767920675142...; in between them we find e^7 = 1096.63315842845859926372... and e^8 = 2980.957987041728274743592... Hence a(6) = 2.
		

Crossrefs

Cf. A000796 (Pi), A001113 (e), A053510 (log(Pi)), A059561 (floor(n*log(Pi))).

Programs

Maple

Digits:= 30:
log_Pi:= evalf(log(Pi));
a:= n-> floor((n+1)*log_Pi) -floor(n*log_Pi):
seq(a(n), n=0..80);  # Alois P. Heinz, Dec 21 2018

Mathematica

Table[Floor[(n + 1)Log[Pi]] - Floor[n Log[Pi]], {n, 0, 99}] (* Alonso del Arte, Dec 21 2018 *)

Scala

val logPi = Math.log(Math.PI); for (n <- 0 to 99) yield (Math.floor(logPi  * (n + 1)) - Math.floor(logPi * n)).toInt // Alonso del Arte, Dec 21 2018

Formula

a(n) = floor((n + 1) log Pi) - floor(n log Pi). - Alonso del Arte, Dec 20 2018