Wolfe Padawer - cp's OEIS frontend

A363089 Positive numbers k for which cos(k) > sin(k).

4, 5, 6, 7, 11, 12, 13, 17, 18, 19, 23, 24, 25, 30, 31, 32, 36, 37, 38, 42, 43, 44, 48, 49, 50, 51, 55, 56, 57, 61, 62, 63, 67, 68, 69, 74, 75, 76, 80, 81, 82, 86, 87, 88, 92, 93, 94, 95, 99, 100, 101, 105, 106, 107, 111, 112, 113, 118, 119, 120, 124, 125, 126, 130, 131, 132

Offset: 1

Wolfe Padawer, May 18 2023

nonn

Terms of this sequence come in groups of 3 or 4 consecutive integers, with spaces between them of length 3 or 4. This is a direct consequence of the fact that 3 < Pi < 4. Across the entire infinite sequence, the percentage of groups of consecutive integers that have 4 members (and the percentage of spaces that are of length 4) is (Pi - 3)*100% = 14.1592653589...%. In the integers between 1 and 10^12, there are 159154943092 groups, of which 22535170724 are length 4, a percentage of 14.1592653587...%, which matches Pi to 11 decimal places.

Positive numbers k such that frac(3/8 + k/(2 * Pi)) < 1/2. - Robert Israel, Jun 18 2024

Robert Israel, Table of n, a(n) for n = 1..10000

Complement of A363088.

filter:= k -> is(frac(k/(2*Pi) + 3/8) < 1/2):
select(filter, [$1..200]); # Robert Israel, Jun 18 2024

A363088 Positive numbers k for which sin(k) >= cos(k).

Original entry on oeis.org

1, 2, 3, 8, 9, 10, 14, 15, 16, 20, 21, 22, 26, 27, 28, 29, 33, 34, 35, 39, 40, 41, 45, 46, 47, 52, 53, 54, 58, 59, 60, 64, 65, 66, 70, 71, 72, 73, 77, 78, 79, 83, 84, 85, 89, 90, 91, 96, 97, 98, 102, 103, 104, 108, 109, 110, 114, 115, 116, 117, 121, 122, 123, 127, 128, 129

Offset: 1

Wolfe Padawer, May 18 2023

nonn

Terms of the sequence come in groups of 3 or 4 consecutive integers, with spaces between them of length 3 or 4. This is a direct consequence of the fact that 3 < Pi < 4. Across the entire infinite sequence, the percentage of groups of consecutive integers that have 4 members (and the percentage of spaces that are of length 4) is (Pi - 3)*100% = 14.1592653589...%. In the integers between 1 and 10^12, there are 159154943092 groups, of which 22535170725 are of length 4, a percentage of 14.1592653594...%, which matches Pi to 10 decimal places.

Complement of A363089.

Select[Range[200], Sin[# - Pi/4] > 0 &] (* Vaclav Kotesovec, Jul 01 2023 *)

A362843 Numbers that are equal to the sum of their digits raised to consecutive odd numbered powers (1,3,5,7,...).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 463, 3943, 371915027434113

Offset: 1

Wolfe Padawer, May 05 2023

Unlike A032799 and A208130, this sequence is not easily proven to be finite. With m >= 1, 10^(m - 1) exceeds 9^1 + 9^2 + ... + 9^m when m is approximately 22.97, meaning it is impossible for an integer with 23 or more digits to be equal to the sum of its digits raised to the consecutive powers. However, 10^(m - 1) will never exceed 9^1 + 9^3 + ... + 9^(2m - 1) over m >= 1. It appears that 10^(m - 1) will never exceed 9^1 + 9^(1 + x) + 9^(1 + 2x) ... 9^(mx - x + 1) over m >= 1 when x >= A154160, approximately 1.04795. For A032799, x = 1, and for this sequence, x = 2. This means this sequence could theoretically be infinite, although it is currently unknown whether it is.

a(14) > 10^24 if it exists. The expected number of k-digit terms can be heuristically estimated as about 10^(-0.15*k), which suggests that the sequence is likely finite. - Max Alekseyev, May 17 2025

			1 = 1^1;
463 = 4^1 + 6^3 + 3^5;
3943 = 3^1 + 9^3 + 4^5 + 3^7.

Cf. A032799, A154160, A208130.

kmax=10^6; a={}; For[k=0, k<=kmax, k++,If[Sum[Part[IntegerDigits[k],i]^(2i-1),{i,IntegerLength[k]}]==k, AppendTo[a,k]]]; a (* Stefano Spezia, May 06 2023 *)

isok(k) = my(d=digits(k)); sum(i=1, #d, d[i]^(2*i-1)) == k; \\ Michel Marcus, May 06 2023

from itertools import count, islice
def A362843_gen(startvalue=0): # generator of terms >= startvalue
    return filter(lambda n:n==sum(int(d)**((i<<1)+1) for i,d in enumerate(str(n))),count(max(startvalue,0)))
A362843_list = list(islice(A362843_gen(),12)) # Chai Wah Wu, Jun 26 2023

a(13) from Martin Ehrenstein, Jul 07 2023

A362219 Decimal expansion of smallest positive solution to tan(x) = arctan(x).

Original entry on oeis.org

4, 0, 6, 7, 5, 8, 8, 8, 6, 5, 7, 6, 5, 8, 6, 2, 7, 9, 0, 9, 1, 7, 0, 8, 5, 0, 2, 5, 3, 1, 2, 4, 1, 1, 3, 1, 9, 0, 6, 8, 3, 0, 0, 6, 7, 4, 4, 9, 3, 9, 5, 7, 9, 2, 2, 6, 3, 7, 2, 6, 3, 4, 3, 6, 5, 5, 1, 4, 6, 5, 8, 6, 2, 6, 6, 0, 5, 4, 7, 1, 0, 1, 5, 5, 9, 0, 2, 8, 2, 3, 7, 7, 0, 4, 4, 0, 0, 1, 1, 6, 8, 2, 0

Offset: 1

Wolfe Padawer, Apr 11 2023

			4.067588865765862790917085025312411319068300674493957922637263436551...

Cf. A115365, A362220.

RealDigits[FindRoot[Tan[x] == ArcTan[x], {x, 4}, WorkingPrecision -> 105][[1, 2]]][[1]]

solve(x=4, 4.5, tan(x)-atan(x)) \\ Michel Marcus, Apr 12 2023

Equals tan(A362220).

A362220 Decimal expansion of smallest positive root of x = tan(tan(x)).

Original entry on oeis.org

1, 3, 2, 9, 7, 3, 1, 2, 2, 0, 6, 7, 8, 9, 4, 5, 5, 1, 5, 7, 3, 7, 1, 4, 6, 0, 6, 5, 5, 8, 4, 6, 4, 8, 5, 8, 9, 6, 0, 4, 8, 2, 9, 8, 5, 7, 4, 9, 0, 3, 8, 0, 4, 3, 6, 7, 5, 1, 2, 4, 6, 4, 5, 7, 9, 7, 9, 9, 7, 8, 0, 4, 7, 0, 6, 0, 1, 4, 3, 2, 0, 4, 5, 8, 3, 8, 2, 3, 7, 1, 3, 6, 9, 5, 1, 6, 2, 4, 8, 8, 4, 3, 6

Offset: 1

Wolfe Padawer, Apr 11 2023

			1.329731220678945515737146065584648589604829857490380436751246457979...

Cf. A115365, A362219.

RealDigits[FindRoot[Tan[Tan[x]] == x, {x, 1.3}, WorkingPrecision -> 105][[1, 2]]][[1]]

solve(x=1.32, 1.35, tan(tan(x)) - x) \\ Michel Marcus, Apr 12 2023

Equals tan(A362219).

Equals arctan(A362219).

A361519 Decimal expansion of arccsch(Pi).

Original entry on oeis.org

3, 1, 3, 1, 6, 5, 8, 8, 0, 4, 5, 0, 8, 6, 8, 3, 7, 5, 8, 7, 1, 8, 6, 9, 3, 0, 8, 2, 6, 5, 7, 0, 5, 9, 3, 1, 5, 2, 5, 1, 4, 2, 0, 4, 5, 5, 2, 0, 2, 1, 0, 1, 0, 5, 6, 4, 1, 5, 1, 0, 7, 0, 6, 4, 7, 2, 5, 8, 1, 9, 7, 3, 7, 6, 3, 4, 8, 0, 7, 5, 2, 2, 8, 5, 6, 2, 5, 3, 8, 0, 9, 8, 6, 0, 6, 7, 8, 0, 5, 0, 1, 8, 6, 6, 0

Offset: 0

Wolfe Padawer, Mar 14 2023

			0.313165880450868375871869308265705931525142045520210105641510706472...

Michael I. Shamos, A catalog of the real numbers (2011).

Cf. A359540, A360938, A361518.

Mathematica
```
RealDigits[ArcCsch[Pi], 10, 105][[1]]
```

asinh(1/Pi) \\ Michel Marcus, Mar 15 2023

Equals arcsinh(1/Pi).
Equals log(sqrt(1/Pi^2 + 1) + 1/Pi).
Equals Integral_{x=Pi..oo} 1/(x*sqrt(1 + x^2)) dx.

A361518 Decimal expansion of arccoth(Pi).

Original entry on oeis.org

3, 2, 9, 7, 6, 5, 3, 1, 4, 9, 5, 6, 6, 9, 9, 1, 0, 7, 6, 1, 7, 8, 6, 3, 4, 1, 7, 5, 5, 5, 2, 1, 8, 6, 0, 4, 2, 7, 0, 1, 3, 7, 3, 9, 1, 1, 4, 0, 6, 9, 2, 4, 1, 4, 4, 0, 2, 9, 0, 8, 3, 5, 4, 7, 6, 2, 0, 0, 6, 2, 8, 3, 7, 3, 1, 5, 6, 7, 1, 7, 2, 8, 6, 1, 1, 8, 2, 6, 3, 6, 4, 8, 6, 3, 6, 2, 7, 1, 4, 0, 8, 0, 1, 6, 5

Offset: 0

Wolfe Padawer, Mar 14 2023

			0.329765314956699107617863417555218604270137391140692414402908354762...

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 31, page 291.

Michael I. Shamos, A catalog of the real numbers (2011).

Cf. A359540, A360938, A361519.

Mathematica
```
RealDigits[ArcCoth[Pi], 10, 105][[1]]
```

atanh(1/Pi) \\ Michel Marcus, Mar 15 2023

Equals arctanh(1/Pi).
Equals (log(1 + 1/Pi) - log(1 - 1/Pi))/2.
Equals Sum_{k>=0} (Pi^(-2k - 1))/(2k + 1).
Equals Integral_{x=1..(1 + 1/Pi)} 1/(2x - x^2) dx.

A360700 Decimal expansion of arcsec(Pi).

Original entry on oeis.org

1, 2, 4, 6, 8, 5, 0, 2, 1, 9, 8, 6, 2, 9, 1, 5, 8, 9, 9, 2, 5, 0, 3, 6, 8, 8, 6, 1, 0, 1, 0, 9, 7, 7, 6, 0, 6, 4, 2, 5, 8, 2, 8, 5, 7, 4, 2, 1, 5, 4, 3, 1, 5, 8, 4, 9, 5, 0, 4, 1, 0, 9, 1, 7, 8, 9, 4, 3, 2, 3, 9, 7, 3, 9, 6, 2, 2, 0, 1, 6, 0, 9, 1, 2, 1, 6, 7, 9, 3, 5, 3, 7, 3, 2, 1, 0, 4, 1, 9, 0, 2, 0, 4, 8, 0

Offset: 1

Wolfe Padawer, Feb 16 2023

			1.246850219862915899250368861010977606425828574215431584950410917894...

Michael I. Shamos, A catalog of the real numbers (2011).

Cf. A232272, A232273, A360701.

Mathematica
```
RealDigits[ArcSec[Pi], 10, 105][[1]]
```

Equals Pi/2 - arccsc(Pi) = Pi/2 - A360701.
Equals arcsin(sqrt(1 - 1/Pi^2)).
Equals arctan(sqrt(Pi^2 - 1)).
Equals |arcsech(Pi)|.
Equals arccos(1/Pi).

A360938 Decimal expansion of arcsinh(Pi).

Original entry on oeis.org

1, 8, 6, 2, 2, 9, 5, 7, 4, 3, 3, 1, 0, 8, 4, 8, 2, 1, 9, 8, 8, 8, 3, 6, 1, 3, 2, 5, 1, 8, 2, 6, 2, 0, 5, 7, 4, 9, 0, 2, 6, 7, 4, 1, 8, 4, 9, 6, 1, 5, 5, 4, 7, 6, 5, 6, 1, 2, 8, 7, 9, 5, 1, 4, 4, 2, 3, 7, 3, 6, 5, 4, 5, 7, 3, 5, 7, 9, 8, 0, 0, 2, 9, 5, 1, 8, 7, 1, 9, 9, 7, 0, 1, 5, 6, 6, 1, 4, 6, 3, 3, 4, 5, 8, 5

Offset: 1

Wolfe Padawer, Feb 26 2023

			1.862295743310848219888361325182620574902674184961554765612879514423...

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 31, page 291.

Cf. A188725, A359540, A361518, A361519.

Mathematica
```
RealDigits[ArcSinh[Pi], 10, 105][[1]]
```
PARI
```
asinh(Pi) \\ Michel Marcus, Feb 26 2023
```

Equals log(Pi + sqrt(Pi^2 + 1)) = log(A188725).

Equals Pi*Integral_{x=0..1} 1/sqrt((Pi^2)*x^2 + 1) dx.

A360701 Decimal expansion of arccsc(Pi).

Original entry on oeis.org

3, 2, 3, 9, 4, 6, 1, 0, 6, 9, 3, 1, 9, 8, 0, 7, 1, 9, 9, 8, 0, 9, 5, 2, 8, 3, 0, 6, 2, 8, 7, 7, 3, 8, 3, 5, 6, 7, 2, 7, 5, 6, 1, 2, 5, 4, 7, 2, 1, 2, 1, 3, 2, 5, 5, 3, 7, 0, 6, 1, 3, 7, 8, 2, 5, 9, 5, 8, 4, 2, 2, 9, 1, 8, 0, 9, 0, 2, 8, 9, 0, 1, 9, 2, 3, 3, 8, 0, 5, 8, 9, 3, 8, 9, 5, 4, 3, 4, 3, 7, 8, 6, 2, 7, 1

Offset: 0

Wolfe Padawer, Feb 16 2023

			0.323946106931980719980952830628773835672756125472121325537061378259...

Michael I. Shamos, A catalog of the real numbers (2011).

Cf. A232272, A232273, A360700.

Mathematica
```
RealDigits[ArcCsc[Pi], 10, 105][[1]]
```

Equals Pi/2 - arcsec(Pi).
Equals arccos(sqrt(1 - 1/Pi^2)).
Equals arctan(1/(sqrt(Pi^2 - 1))).

cp's OEIS Frontend

User: Wolfe Padawer

A363089 Positive numbers k for which cos(k) > sin(k).

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Maple

A363088 Positive numbers k for which sin(k) >= cos(k).

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Mathematica

A362843 Numbers that are equal to the sum of their digits raised to consecutive odd numbered powers (1,3,5,7,...).

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Extensions

A362219 Decimal expansion of smallest positive solution to tan(x) = arctan(x).

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A362220 Decimal expansion of smallest positive root of x = tan(tan(x)).

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Mathematica

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A361519 Decimal expansion of arccsch(Pi).

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A361518 Decimal expansion of arccoth(Pi).

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A360700 Decimal expansion of arcsec(Pi).

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