Mark Andreas - cp's OEIS frontend

A360080 Smallest k such that 2^(2^n) + k is a safe prime.

1, 7, 7, 7, 91, 3103, 12451, 230191, 286867, 1657867, 10029811, 29761351, 22410151, 98402791, 167137543

Offset: 1

Mark Andreas, Jan 25 2023

a(n) == 3 (mod 4) for n > 1. - Chai Wah Wu, Jan 27 2023

			a(3) = 7 because 2^(2^3) + 7 = 263 is the smallest safe prime greater than 256.

Cf. A360081, A181356, A005385, A058220, A013597, A013603, A335313, A350696.

a(n) = {my(k=1); pow2 = 2^(2^n);   while (!(isprime(pow2 + k) && isprime((pow2 + k - 1)/2)), k+=2); k;} \\

from sympy import isprime, nextprime
def A360080(n):
    if n <= 1: return 1
    m = 1<<(1<Chai Wah Wu, Jan 27 2023

a(n) = A350696(2^n).

A360081 Smallest k such that 2^(3*2^n) + k is a safe prime.

Original entry on oeis.org

3, 19, 31, 691, 907, 2887, 15943, 69283, 216127, 1108831, 8344423, 10976347, 166965391, 385465771, 26580643

Offset: 0

Mark Andreas, Jan 25 2023

a(n) == 3 (mod 4). - Chai Wah Wu, Jan 27 2023

			a(1) = 19 because 2^(3*2^1)+19 = 2^6+19 = 83 is the smallest safe prime greater than 64.

Cf. A360080, A335313, A350696, A005385, A013597, A013603, A181356, A057821.

a(n) = {my(k=1); pow2 = 2^(3*2^n); while (!(isprime(pow2 + k) && isprime((pow2 + k - 1)/2)), k+=2); k;} \\

from sympy import isprime, nextprime
def A360081(n):
    m = 1<<3*(1<Chai Wah Wu, Jan 27 2023

a(n) = A350696(3*2^n).

A351217 Decimal expansion of the 20th root of 3.

Original entry on oeis.org

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

Offset: 1

Mark Andreas, Apr 22 2022

			1.0564673085495378613933514529880342628505731...

Cf. A351212 (15th root), A351213 (16th root), A351214 (17th root), A351215 (18th root), A351216 (19th root).

Maple
```
Digits:=100: evalf(3^(1/20));
```

RealDigits[3^(1/20), 10, 108][[1]]
RealDigits[Surd[3,20],10,120][[1]] (* Harvey P. Dale, May 28 2025 *)

PARI
```
sqrtn(3,20)
```

Equals 3^(1/20) = A246711^(1/2).

A351216 Decimal expansion of the 19th root of 3.

Original entry on oeis.org

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

Offset: 1

Mark Andreas, Apr 22 2022

			1.0595260647382752026415391804852171711346969...

Cf. A351211 (14th root), A351212 (15th root), A351213 (16th root), A351214 (17th root), A351215 (18th root).

Maple
```
Digits:=100: evalf(3^(1/19));
```
Mathematica
```
RealDigits[3^(1/19), 10, 108][[1]]
```
PARI
```
sqrtn(3,19)
```

Equals 3^(1/19).

A351215 Decimal expansion of the 18th root of 3.

Original entry on oeis.org

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

Offset: 1

Mark Andreas, Apr 22 2022

			1.0629350704110543330004157448697337637113387...

Cf. A351210 (13th root), A351211 (14th root), A351212 (15th root), A351213 (16th root), A351214 (17th root).

Maple
```
Digits:=100: evalf(3^(1/18));
```
Mathematica
```
RealDigits[3^(1/18), 10, 108][[1]]
```
PARI
```
sqrtn(3,18)
```

Equals 3^(1/18).

A351214 Decimal expansion of the 17th root of 3.

Original entry on oeis.org

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

Offset: 1

Mark Andreas, Mar 09 2022

			1.0667581171328452951068613563212343784040253...

Cf. A351209 (12th root) A351210 (13th root) A351211 (14th root) A351212 (15th root) A351212 (16th root).

Maple
```
Digits:=100: evalf(3^(1/17));
```
Mathematica
```
RealDigits[3^(1/17), 10, 108][[1]]
```
PARI
```
sqrtn(3,17)
```

Equals 3^(1/17).

A351213 Decimal expansion of the 16th root of 3.

Original entry on oeis.org

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

Offset: 1

Mark Andreas, Mar 09 2022

			1.071075483072914469123247...

Cf. A351208 (11th root), A351209 (12th root), A351210 (13th root), A351211 (14th root), A351212 (15th root).

Maple
```
Digits:=100: evalf(3^(1/16));
```
Mathematica
```
RealDigits[3^(1/16), 10, 108][[1]]
```
PARI
```
sqrtn(3,16)
```

Equals 3^(1/16).

A351212 Decimal expansion of the 15th root of 3.

Original entry on oeis.org

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

Offset: 1

Mark Andreas, Mar 09 2022

			1.0759896247253458666946663627239770760895246...

Cf. A351208 (11th root), A351209 (12th root), A351210 (13th root), A351211 (14th root).

Maple
```
Digits:=100: evalf(3^(1/15));
```
Mathematica
```
RealDigits[3^(1/15), 10, 108][[1]]
```
PARI
```
sqrtn(3,15)
```

Equals 3^(1/15).

A351211 Decimal expansion of the 14th root of 3.

Original entry on oeis.org

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

Offset: 1

Mark Andreas, Feb 07 2022

			1.0816334003527659585671688318749777839374638...

Index entries for algebraic numbers, degree 14.

Cf. A246711 (10th root), A351208 (11th root), A351209 (12th root), A351209 (13th root).

Maple
```
Digits:=100: evalf(3^(1/14));
```
Mathematica
```
RealDigits[3^(1/14), 10, 108][[1]]
```
PARI
```
sqrtn(3,14)
```

from sympy import integer_nthroot
def A351211(n): return integer_nthroot(3*10**(14*(n-1)),14)[0] % 10 # Chai Wah Wu, Mar 07 2022

Equals 3^(1/14).

A351210 Decimal expansion of the 13th root of 3.

Original entry on oeis.org

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

Offset: 1

Mark Andreas, Feb 07 2022

			1.0881822434633167374809461526062579626660268...

Cf. A011446 (9th root), A246711 (10th root), A351208 (11th root), A351209 (12th root).

Maple
```
Digits:=100: evalf(3^(1/13));
```
Mathematica
```
RealDigits[3^(1/13), 10, 108][[1]]
```
PARI
```
sqrtn(3,13)
```

from sympy import integer_nthroot
def A351210(n): return integer_nthroot(3*10**(13*(n-1)),13)[0] % 10 # Chai Wah Wu, Mar 07 2022

Equals 3^(1/13).

cp's OEIS Frontend

User: Mark Andreas

A360080 Smallest k such that 2^(2^n) + k is a safe prime.

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A360081 Smallest k such that 2^(3*2^n) + k is a safe prime.

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Python

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A351217 Decimal expansion of the 20th root of 3.

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A351216 Decimal expansion of the 19th root of 3.

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A351215 Decimal expansion of the 18th root of 3.

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A351214 Decimal expansion of the 17th root of 3.

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A351213 Decimal expansion of the 16th root of 3.

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Maple

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Formula

A351212 Decimal expansion of the 15th root of 3.