Patrick C. Schneider - cp's OEIS frontend

A305274 Decimal expansion of x such that x^(x+3) = x^x + x^3.

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

Offset: 1

Patrick C. Schneider, Aug 18 2018

			1.393091545385710592226140783294461940981161317845356960802924...

Cf. A008789 (n^(n+3)), A316295 (similar, with 2 instead of 3).

RealDigits[x /. FindRoot[x^(x + 3) == x^x + x^3, {x, 3/2}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Jun 18 2023 *)

PARI
```
solve(x=1,2,x^(x+3)-x^x-x^3)
```

A319026 Decimal expansion of Psi(3).

Original entry on oeis.org

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

Offset: 0

Patrick C. Schneider, Sep 08 2018

Psi(x) is the digamma function (logarithmic derivative of the Gamma function).

			0.92278433509846713939348790...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, pp. 258-259 [alternative scanned copy].

Cf. A001620, A153810.

evalf(Psi(3.0)) ; - R. J. Mathar, Aug 29 2023

RealDigits[3/2 - EulerGamma, 10, 100][[1]] (* Amiram Eldar, May 24 2023 *)

PARI
```
psi(3)
```

Psi(3) = Psi(2) + 1/2 = 3/2 - gamma, with gamma = A001620 and Psi(2) = A153810. - Wolfdieter Lang, Oct 05 2018

A316295 Decimal expansion of x such that x^(x+2) = x^x + x^2.

Original entry on oeis.org

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

Offset: 1

Patrick C. Schneider, Jun 29 2018

			1.4917797093328498412523283387465897735355953027256882082581554376779776...

Cf. A008788 (x^(x+2)).

RealDigits[ NSolve[x^(x + 2) == x^x + x^2, x, Reals, WorkingPrecision -> 128][[1, 1, 2]], 10][[1]] (* Robert G. Wilson v, Jul 21 2018 *)

PARI
```
solve(x=1, 2, x^(x+2)-x^x-x^2)
```

A316360 Numbers k such that (2^k-2)^2-3 is prime.

Original entry on oeis.org

4, 8, 12, 14, 18, 32, 62, 66, 96, 120, 122, 140, 180, 202, 228, 740, 800, 810, 1012, 1142, 1386, 1496, 1698, 4622, 5674, 54892, 75482

Offset: 1

Patrick C. Schneider, Jun 30 2018

[n: n in [2..750] |IsPrime((2^n - 2)^2 - 3)]; // Vincenzo Librandi, Jul 01 2018

select(k->isprime(4^k-2^(2+k)+1),[$ 1..3000]); # Muniru A Asiru, Jul 01 2018

Select[Range[3000], PrimeQ[(2^# - 2)^2 - 3] &] (* Michael De Vlieger, Jun 30 2018 *)

PARI
```
is(n)=ispseudoprime((2^n-2)^2-3)
```

a(26) from Giovanni Resta, Jul 02 2018

a(27) from Michael S. Branicky, Nov 27 2024

A316229 Decimal expansion of x such that x^x - x^-x = x.

Original entry on oeis.org

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

Offset: 1

Patrick C. Schneider, Jun 27 2018

			1.5819790225351794653791459960674390425384...

Patrick C. Schneider, Table of n, a(n) for n = 1..10000

Cf. A000312 (x^x).

RealDigits[x /. FindRoot[x^x - x^(-x) == x, {x, 1}, WorkingPrecision -> 100]][[1]] (* Vaclav Kotesovec, Jul 06 2018 *)

PARI
```
solve(x=1, 2, x^x-x^-x-x)
```

A305836 Numbers k such that (2^k-1)^4 + 2 is prime.

Original entry on oeis.org

0, 1, 2, 4, 6, 10, 12, 30, 78, 244, 490, 760, 1808, 4790, 34330

Offset: 1

Patrick C. Schneider, Jun 11 2018

a(16) > 10^5. - Robert Price, Jul 21 2018

Cf. A128832 ((2^n-1)^4).

Select[Range@ 2000, PrimeQ[(2^# - 1)^4 + 2] &] (* Michael De Vlieger, Jun 11 2018 *)

PARI
```
is(n)=ispseudoprime((2^n-1)^4+2)
```

a(14) from Vaclav Kotesovec, Jun 16 2018

a(15) from Robert Price, Jun 20 2018

cp's OEIS Frontend

User: Patrick C. Schneider

A305274 Decimal expansion of x such that x^(x+3) = x^x + x^3.

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A319026 Decimal expansion of Psi(3).

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A316295 Decimal expansion of x such that x^(x+2) = x^x + x^2.

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A316360 Numbers k such that (2^k-2)^2-3 is prime.

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Magma

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Extensions

A316229 Decimal expansion of x such that x^x - x^-x = x.

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Examples

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Mathematica

PARI

A305836 Numbers k such that (2^k-1)^4 + 2 is prime.

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Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Extensions