A121854 -id:A121854 - cp's OEIS frontend

A121901 a(n) = floor((Pi + e)*sqrt(n)).

0, 5, 8, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 21, 22, 23, 24, 24, 25, 26, 26, 27, 28, 28, 29, 29, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35, 36, 36, 37, 37, 37, 38, 38, 39, 39, 40, 40, 41, 41, 41, 42, 42, 43, 43, 43, 44, 44, 45, 45, 45, 46, 46, 46, 47, 47, 47, 48, 48

Offset: 0

Mohammad K. Azarian, Sep 01 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..999 [Offset shifted by _Georg Fischer_, Sep 02 2022]

Cf. A059742 (Pi+e), A121854, A121855, A121856, A121857, A121900, A121902.

C := ComplexField(); [Floor((Exp(1) + Pi(C))*Sqrt(n)): n in [0..50]]; // G. C. Greubel, Oct 28 2017

Mathematica

Table[Floor[(Pi + E) Sqrt[n]], {n, 0, 70}] (* Vincenzo Librandi, Feb 21 2013 *)

PARI

for(n=0,50, print1(floor((exp(1)+Pi)*sqrt(n)), ", ")) \\ G. C. Greubel, Oct 28 2017

Extensions

Offset corrected by Mohammad K. Azarian, Nov 20 2008

Offset changed to 0 by Georg Fischer, Sep 02 2022

A121902 a(n) = floor((Pi - e)*sqrt(n)).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 0

Mohammad K. Azarian, Sep 01 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..999 [Offset shifted by _Georg Fischer_, Sep 02 2022]

Cf. A073244 (Pi-e), A121854, A121855, A121856, A121857, A121899, A121900, A121901.

C := ComplexField(); [Floor((Pi(C) - Exp(1))*Sqrt(n)): n in [0..50]]; // G. C. Greubel, Oct 28 2017

Mathematica

Table[Floor[(Pi - E)*Sqrt[n]], {n, 0, 110}] (* Vincenzo Librandi, Feb 21 2013 *)

PARI

for(n=0,50, print1(floor((Pi - exp(1))*sqrt(n)), ", ")) \\ G. C. Greubel, Oct 28 2017

Extensions

Offset corrected by Mohammad K. Azarian, Nov 20 2008

Offset changed to 0 by Georg Fischer, Sep 02 2022

A121903 a(n) = ceiling(Pi^(n*e)).

Original entry on oeis.org

1, 23, 505, 11329, 254434, 5714357, 128339633, 2882400038, 64736277049, 1453922256330, 32653869265130, 733378399940297, 16471061151498381, 369926160190271627, 8308229975861003526, 186595847388277259848, 4190785566084546949288, 94121513992523815815370

Offset: 0

Mohammad K. Azarian, Sep 01 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..699

Cf. A121854, A121855, A121856, A121857, A121904.

C := ComplexField(); [Ceiling(Pi(C)^(n*Exp(1))): n in [0..50]]; // G. C. Greubel, Nov 06 2017

Mathematica

Ceiling[Pi^(E Range[0, 20])] (* Vincenzo Librandi, Feb 21 2013 *)

PARI

for(n=0,50, print1(ceil(Pi^(n*exp(1))), ", ")) \\ G. C. Greubel, Nov 06 2017

Extensions

Offset corrected by Georg Fischer, Aug 18 2022

Offset corrected for b-file by Sidney Cadot, Jan 06 2023

A121916 a(n) = floor((Pi+e)^(n*e)).

Original entry on oeis.org

1, 122, 14951, 1828144, 223535959, 27332807665, 3342112728281, 408656059975457, 49968325108097955, 6109865382293662597, 747082374864324679924, 91349324397617876090443, 11169717488538903806777417, 1365774619533204572560235117

Offset: 0

Mohammad K. Azarian, Sep 02 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..399 [Offset shifted by _Georg Fischer_, Sep 02 2022]

Cf. A121854, A121855, A121856, A121857, A121915.

C := ComplexField(); [Floor((Pi(C) + Exp(1))^(n*Exp(1))): n in [0..50]]; // G. C. Greubel, Nov 06 2017

Mathematica

Floor[(Pi + E)^(E (Range[0, 20]))] (* Vincenzo Librandi, Feb 21 2013 *)

PARI

for(n=0,50, print1(floor((Pi+exp(1))^(n*exp(1))), ", ")) \\ G. C. Greubel, Nov 06 2017

Extensions

Offset changed to 0 by Georg Fischer, Sep 02 2022

A121918 a(n) = ceiling((Pi+e)^(n*Pi)).

Original entry on oeis.org

1, 259, 66802, 17265409, 4462406596, 1153350806022, 298094324981779, 77045272021641917, 19913072619720776033, 5146720243221262934094, 1330218081751512472685764, 343807329988307215923432747, 88860226586342124489251555257, 22966758356328845813340839281382

Offset: 0

Mohammad K. Azarian, Sep 02 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..199 [Offset shifted by _Georg Fischer_, Sep 02 2022]

Cf. A121854, A121855, A121856, A121857, A121915, A121917.

C := ComplexField(); [Ceiling((Pi(C) + Exp(1))^(n*Pi(C))): n in [0..50]]; // G. C. Greubel, Nov 06 2017

Mathematica

With[{a = \[Pi] + E}, Ceiling[a^(\[Pi] Range[0, 20])]] (* Vincenzo Librandi, Feb 21 2013 *)

PARI

for(n=0,50, print1(ceil((Pi+exp(1))^(n*Pi)), ", ")) \\ G. C. Greubel, Nov 06 2017

Extensions

Offset changed to 0 by Georg Fischer, Sep 02 2022

A121928 a(n) = ceiling(n*(e^Pi - Pi^e)).

Original entry on oeis.org

0, 1, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8, 9, 9, 10, 11, 11, 12, 13, 13, 14, 15, 15, 16, 17, 18, 18, 19, 20, 20, 21, 22, 22, 23, 24, 24, 25, 26, 26, 27, 28, 28, 29, 30, 30, 31, 32, 33, 33, 34, 35, 35, 36, 37, 37, 38, 39, 39, 40, 41, 41, 42, 43, 43, 44, 45, 45, 46, 47, 48, 48, 49, 50, 50

Offset: 0

Mohammad K. Azarian, Sep 02 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..999 [Offset shifted by _Georg Fischer_, Sep 02 2022]

Cf. A121854, A121855, A121856, A121857, A121929, A121930.

C := ComplexField(); [Ceiling(n*(Exp(1)^Pi(C) - Pi(C)^Exp(1) )): n in [0..50]]; // G. C. Greubel, Nov 06 2017

Mathematica

With[{a = E^Pi - Pi^E}, Ceiling[a Range[0, 80]]] (* Vincenzo Librandi, Feb 21 2013 *)

PARI

for(n=0,50, print1(ceil(n*(exp(Pi) - Pi^exp(1))), ", ")) \\ G. C. Greubel, Nov 06 2017

Extensions

Offset corrected by Mohammad K. Azarian, Nov 20 2008

Offset changed to 0 by Georg Fischer, Sep 02 2022

A121925 a(n) = floor(n*(Pi^e + e^Pi)).

Original entry on oeis.org

0, 45, 91, 136, 182, 227, 273, 319, 364, 410, 455, 501, 547, 592, 638, 683, 729, 775, 820, 866, 911, 957, 1003, 1048, 1094, 1139, 1185, 1231, 1276, 1322, 1367, 1413, 1459, 1504, 1550, 1595, 1641, 1687, 1732, 1778, 1823, 1869, 1915, 1960, 2006, 2051, 2097

Offset: 0

Mohammad K. Azarian, Sep 02 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..999 [Offset shifted by _Georg Fischer_, Sep 02 2022]

Cf. A121854, A121855, A121856, A121857, A121929.

C := ComplexField(); [Floor(n*(Pi(C)^Exp(1) + Exp(1)^Pi(C))): n in [0..50]]; // G. C. Greubel, Nov 06 2017

Mathematica

With[{a = Pi^E + E^Pi}, Floor[a * Range[0, 40]]] (* Vincenzo Librandi, Feb 20 2013 *)

PARI

a(n)=floor(n*(Pi^exp(1)+exp(Pi))) \\ Charles R Greathouse IV, Feb 20 2013

Extensions

Offset corrected by Mohammad K. Azarian, Nov 20 2008

Offset changed to 0 by Georg Fischer, Sep 02 2022

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

Original entry on oeis.org

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

Jianing Song, Feb 05 2020

nonn

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).

			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.

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)

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

Cf. A331859, A002486, A002485.

Magma
```
// See Schoenfield link.
```

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

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

cp's OEIS Frontend

A121901 a(n) = floor((Pi + e)*sqrt(n)).

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Magma

Mathematica

PARI

Extensions

A121902 a(n) = floor((Pi - e)*sqrt(n)).

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Magma

Mathematica

PARI

Extensions

A121903 a(n) = ceiling(Pi^(n*e)).

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Programs

Magma

Mathematica

PARI

Extensions

A121916 a(n) = floor((Pi+e)^(n*e)).

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Magma

Mathematica

PARI

Extensions

A121918 a(n) = ceiling((Pi+e)^(n*Pi)).

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Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Extensions

A121928 a(n) = ceiling(n*(e^Pi - Pi^e)).

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Author

Keywords

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Crossrefs

Programs

Magma

Mathematica

PARI

Extensions

A121925 a(n) = floor(n*(Pi^e + e^Pi)).

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Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Extensions

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

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