Christoph B. Kassir - cp's OEIS frontend

A359278 Antidiagonal sums of A354967.

1, 4, 9, 19, 45, 127, 491, 2597, 18905, 180253, 2176101, 32236029, 571516361, 11885151437, 285237145381, 7794999370341, 239867327549943, 8232788441242931, 312611538663794793, 13040004273788572983

Offset: 1

Christoph B. Kassir, Feb 03 2023

Cf. A000040, A058009, A354967, A345669.

a(n) = sum(k=1, n, my(p=k); for(j=k+1, n, p=prime(p)); p) \\ Andrew Howroyd, Feb 04 2023

from sympy import prime
def nprime(x, y):
    p = x
    for _ in range(y):
        p = prime(p)
    return p
def a(n):
    return sum([nprime(k, n-k) for k in range(1, n+1)])

a(n) = Sum_{k=1..n} p(k, n-k+1), where p(n,0) = n and p(n,k) = prime(p(n, k-1)) for k >= 1.

a(9)-a(15) from Andrew Howroyd, Feb 03 2023

a(16)-a(20) from Alois P. Heinz, Feb 03 2023

A356983 Decimal expansion of Pi * e^(-Pi/2).

Original entry on oeis.org

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

Offset: 0

Christoph B. Kassir, Sep 07 2022

			0.653072949894912131389531881117225431...

Michael Penn, This integral looks crazy, YouTube video, 2022.

Cf. A000796, A001113, A049006.

RealDigits[Pi * Exp[-Pi/2], 10, 100][[1]]

PARI
```
Pi * exp(-Pi/2)
```

Equals Integral_{x=0..Pi} i^tan(x) dx, where i is the imaginary unit.

A356571 a(n) = floor(f(n)), where f(n) = n^4(15-24n+10n^2) + 20n^5(1-n)^3 / (1-2n(1-n)).

Original entry on oeis.org

0, 1, -16, -318, -1895, -6936, -19313, -45055, -92831, -174433, -305249, -504751, -796967, -1210969, -1781345, -2548687, -3560063, -4869505, -6538481, -8636383, -11240999, -14439001, -18326417, -23009119, -28603295, -35235937, -43045313, -52181455, -62806631, -75095833, -89237249

Offset: 0

Christoph B. Kassir, Aug 12 2022

sign

Sandefur shows that if the probability of winning any particular point in a tennis match is p, the fraction of the games won would be f(p).

James Sandefur, A Geometric Series from Tennis, The College Mathematics Journal, Vol. 36, No. 3 (May, 2005), pp. 224-226.

a[n_] := Floor[n^4*(15 - 24*n + 10 n^2) + 20*n^5*(1 - n)^3/(1 - 2*n*(1 - n))]; Array[a, 30, 0] (* Amiram Eldar, Aug 12 2022 *)

def a(n):
    return n**4 * (15-24*n+10*n**2) + 20*n**5 * (1-n)**3 // (1-2*n*(1-n))

a(n) = g(n) + h(n), where g(n) = floor(n^2 * (-4*n^3 + 10*n^2 - 5*n - 5/2) + 1) and h(n) = [1 if n=0; 2 if n=1; -1 if n=3,5,6; 0 if n=4 or n>6]

A354077 Decimal expansion of Pi/2 + sqrt(3) - 1.

Original entry on oeis.org

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

Offset: 1

Christoph B. Kassir, Aug 21 2022

This is the perimeter of a single piece in the shape presented in Talwalkar's video.

			2.302847134363773912758768033145623809041...

Presh Talwalkar, What is the perimeter of a single shape?, YouTube video, 2022.
Index entries for transcendental numbers

Cf. A000796, A019669, A002194.

RealDigits[(Pi/2)+Sqrt[3]-1, 10, 120][[1]]

fpprec: 100$ ev(bfloat((%pi/2)+sqrt(3)-1));

PARI
```
(Pi/2)+sqrt(3)-1
```

Equals A019669 + A002194 - 1.

A356418 Decimal expansion of sqrt(4/3 + 1/sqrt(3)).

Original entry on oeis.org

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

Offset: 1

Christoph B. Kassir, Aug 21 2022

This is the radius of the smallest circle containing a system of three unit squares which intersect at the vertices to form an equilateral triangle at the center.

			1.382274792696068482365108044918041903951415152171813...

Presh Talwalkar, Very few people solved this test problem from Australia, YouTube video, 2022.

Cf. A002194, A020760.

Magma
```
Sqrt(4/3 + 1/Sqrt(3));
```
Maple
```
evalf(sqrt(4/3 + 1/sqrt(3)), 120);
```

RealDigits[Sqrt[4/3 + 1/Sqrt[3]], 10, 120][[1]]

fpprec: 100$ ev(bfloat(sqrt(4/3 + 1/sqrt(3))));

PARI
```
sqrt(4/3 + 1/sqrt(3))
```

A356593 Smallest k such that primorial(k) > n^2.

Original entry on oeis.org

1, 2, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 1

Christoph B. Kassir, Aug 14 2022

nonn

Cf. A337769, A002110, A000290.

a[n_] := Module[{k = 1, prod = p = 2}, While[prod < n^2, p = NextPrime[p]; prod *= p; k++]; k]; Array[a, 100] (* Amiram Eldar, Aug 15 2022 *)

from sympy import primorial
def a(n):
    k = 1
    while True:
        if primorial(k) > n**2:
            return(k)
        k += 1
for n in range(1, 90):
    print(f'{a(n)}, ', end='')

A356479 Decimal expansion of (sqrt(3)/Pi) * sinh(Pi/sqrt(3)).

Original entry on oeis.org

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

Offset: 1

Christoph B. Kassir, Aug 08 2022

			1.6459025152253961193544118815663276416192231065463...

Michael Ian Shamos, Shamos's catalog of the real numbers, (2011) [page 520, but misprint in the product 3*k should be 3*k^2].

Cf. A000796, A002194, A334401.

RealDigits[Sqrt[3]*Sinh[Pi/Sqrt[3]]/Pi, 10, 100][[1]] (* Amiram Eldar, Aug 09 2022 *)

PARI
```
print((sqrt(3)/Pi) * sinh(Pi/sqrt(3)))
```

Equals Product_{k>=1} (1 + 1/(3*k^2)).

A356476 Decimal expansion of Loschmidt constant in m^-3 (273.15 K, 100 kPa).

Original entry on oeis.org

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

Offset: 26

Christoph B. Kassir, Aug 08 2022

Has period 1910112. - Jianing Song, Sep 18 2022

			2.651645804 * 10^25 m^-3.

NIST, Loschmidt constant in m^-3 (273.15 K, 100 kPa)
Wikipedia, Loschmidt constant

Cf. A228163, A070063, A070064, A322578.

Equals 10^5 / (273.15 * A070063) = (2/7542485487) * 10^35. - Jianing Song, Sep 18 2022

More terms added by Jianing Song, Sep 18 2022

A355703 a(n) = binomial(n, floor(log(n))).

Original entry on oeis.org

1, 1, 3, 4, 5, 6, 7, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 1330, 1540, 1771, 2024, 2300, 2600, 2925, 3276, 3654, 4060, 4495, 4960, 5456, 5984, 6545, 7140, 7770, 8436, 9139, 9880, 10660, 11480, 12341, 13244, 14190, 15180, 16215, 17296, 18424, 19600, 20825

Offset: 1

Christoph B. Kassir, Jul 14 2022

nonn

Mathematics Stack Exchange, How to compute asymptotic growth of binomial(n, log(n)).

Cf. A000195, A007318.

Cf. A001405, A051033, A051036, A051052, A051053.

a:= n-> binomial(n, ilog(n)):
seq(a(n), n=1..60);  # Alois P. Heinz, Jul 31 2022

a[n_] := Binomial[n, Floor[Log[n]]]; Array[a, 50] (* Amiram Eldar, Jul 31 2022 *)

a(n) = binomial(n, floor(log(n))); \\ Michel Marcus, Jul 31 2022

from numpy import log
from math import comb, floor
for n in range(1, 50):
    x = comb(n, floor(log(n)))
    print("{}, ".format(x), end='')

A356381 (Negated) Decimal expansion of value of absolute zero in degrees Celsius.

Original entry on oeis.org

2, 7, 3, 1, 5

Offset: 3

Christoph B. Kassir, Aug 04 2022

			Absolute zero is exactly 0 Kelvin, exactly -273.15 degrees Celsius, and exactly -459.67 degrees Fahrenheit (A356509).

C. P. Aurora, Thermodynamics, McGraw-Hill Education, 2001, p. 43, Table 2.4.

Cf. A356509 (in Fahrenheit).

cp's OEIS Frontend

User: Christoph B. Kassir

A359278 Antidiagonal sums of A354967.

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A356983 Decimal expansion of Pi * e^(-Pi/2).

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A356571 a(n) = floor(f(n)), where f(n) = n^4*(15-24*n+10*n^2) + 20*n^5*(1-n)^3 / (1-2*n(1-n)).

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A354077 Decimal expansion of Pi/2 + sqrt(3) - 1.

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Mathematica

Maxima

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A356418 Decimal expansion of sqrt(4/3 + 1/sqrt(3)).

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Magma

Maple

Mathematica

Maxima

PARI

A356593 Smallest k such that primorial(k) > n^2.

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Mathematica

Python

A356479 Decimal expansion of (sqrt(3)/Pi) * sinh(Pi/sqrt(3)).

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Examples

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Programs

Mathematica

PARI

Formula

A356476 Decimal expansion of Loschmidt constant in m^-3 (273.15 K, 100 kPa).

A356571 a(n) = floor(f(n)), where f(n) = n^4(15-24n+10n^2) + 20n^5(1-n)^3 / (1-2n(1-n)).