Stefano Spezia - cp's OEIS frontend

A387247 Decimal expansion of (2*log(3) + 7)/8.

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

Offset: 1

Stefano Spezia, Aug 24 2025

			1.149653072167027422848811309230631426161872639...

W. Lai, H. Jiang, D. Zhu, and B. Zhu, Improved Approximation Algorithm and Hardness Result for Sorting Unsigned Strings by Symmetric Reversals. In: Fomin, F.V., Xiao, M. (eds) Computing and Combinatorics. COCOON 2025. Lecture Notes in Computer Science, vol 15984. Springer, Singapore (2026).
Index entries for transcendental numbers.

Cf. A002391, A016632.

Mathematica
```
RealDigits[(2Log[3]+7)/8,10,100][[1]]
```

A387235 Decimal expansion of 2*log(2)/3.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Aug 23 2025

Area enclosed by the curve of the equation x^6 + y^6 - x^3*y + x*y^3 = 0.

The asymptotic mean of A256232. - Amiram Eldar, Aug 23 2025

			0.46209812037329687294482141430545104538366675624...

Michael I. Shamos, A catalog of the real numbers, (2007). See p. 436.
Index entries for transcendental numbers.

Cf. A002162, A001504, A010701, A010722, A016627, A193535, A256232.

Mathematica
```
RealDigits[2Log[2]/3,10,100][[1]]
```

Equals A002162*A010722.
Equals log(4)/3 = A010701*A016627.
Equals Sum_{k>=0} (-1)^k/((3*k + 1)*(3*k + 2)) = Integral_{x=0..1} x^2*log(1 + 1/x^3) = -Integral_{x=0..1} log[1 - x^6]/x^4. [Shamos]
Equals A016627/3 = 2*A193535. - Hugo Pfoertner, Aug 23 2025

A387131 Decimal expansion of the real part of the complex solutions to log(z) = -1/z on the principal branch of log(z).

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Aug 17 2025

			0.16837637908722291055702904019641979887737474829...

Index entries for transcendental numbers.

Cf. A030797, A059526, A059527.

Cf. A387132 (absolute value of the imaginary part).

RealDigits[Re[-1/ProductLog[-1]],10,100][[1]]

Equals Re(-1/LambertW(-1)).

A387132 Decimal expansion of the absolute value of the imaginary part of the complex solutions to log(z) = -1/z on the principal branch of log(z).

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Aug 17 2025

			0.7077541887847276164728458930765162394938408142...

Index entries for transcendental numbers.

Cf. A030797, A059526, A059527.

Cf. A387131 (real part).

RealDigits[Im[-1/ProductLog[-1]],10,100][[1]]

Equals Im(-1/LambertW(-1)).

Equals -Im(-1/LambertW(-1, -1)).

A387101 Decimal expansion of the smallest real solution to e^x = x^3.

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Aug 16 2025

Equivalently, the smallest real solution to log(x) = x/3.

			1.85718386020783533645698098206276669990441533...

Index entries for transcendental numbers.

Cf. A030178, A126583, A126584, A387102 (largest).

RealDigits[-3*ProductLog[-1/3],10,100][[1]]

-3*lambertw(-1/3) \\ Michel Marcus, Aug 18 2025

Equals -3*LambertW(-1/3).

A387102 Decimal expansion of the largest real solution to e^x = x^3.

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Aug 16 2025

Equivalently, the largest real solution to log(x) = x/3.

			4.5364036549735274216902190342161161138109511554...

Index entries for transcendental numbers.

Cf. A030178, A126583, A126584, A366565, A387101 (smallest).

RealDigits[-3*ProductLog[-1,-1/3],10,100][[1]]

Equals -3*LambertW(-1, -1/3).

A386974 a(n) is the permanent of the symmetric n X n matrix M_n where M_n(j,k) = n for j = k, M_n(j,n) = n-j, M_n(n,k) = n-k, M_n(j,k) = 0 otherwise.

Original entry on oeis.org

1, 1, 5, 42, 480, 6875, 117936, 2352980, 53477376, 1363146165, 38500000000, 1193121531646, 40246286745600, 1467779362716303, 57544321060925440, 2413281884765625000, 107798160680740192256, 5109425146945021190505, 256115971082717276995584, 13536555538728461399269330

Offset: 0

Stefano Spezia, Aug 11 2025

nonn

			a(5) = permanent(M_5) = 6875 where M_5 is the matrix
  [5 0 0 0 4]
  [0 5 0 0 3]
  [0 0 5 0 2]
  [0 0 0 5 1]
  [4 3 2 1 5]

Cf. A174963 (determinants), A386975.

M[j_, k_, n_]:=If[j==k, n, If[k==n, n-j, If[j==n, n-k, 0]]]; a[n_]:=Permanent[Table[M[i, j, n], {i, n}, {j, n}]];Join[{1}, Array[a, 18]]

a(n) = matpermanent(matrix(n, n, j, k, if (j==k, n, if (k==n, n-j, if (j==n, n-k, 0))))); \\ Michel Marcus, Aug 12 2025

A386975 a(n) is the permanent of the n X n matrix M_n with M_n(j,k) = j for j <> k, M_n(j,k) = n+j for j = k.

Original entry on oeis.org

1, 2, 14, 183, 3792, 114780, 4807728, 267380071, 19098388480, 1705287529422, 186174804704000, 24402257980061599, 3781731531452940288, 684046276855242721368, 142823583210894978115584, 34092816821609506532859375, 9226267072346511233190461440, 2809774286001810901571097532538

Offset: 0

Stefano Spezia, Aug 11 2025

nonn

			a(5) = permanent(M_5) = 114780 where M_5 is the matrix
  [6, 1, 1, 1,  1]
  [2, 7, 2, 2,  2]
  [3, 3, 8, 3,  3]
  [4, 4, 4, 9,  4]
  [5, 5, 5, 5, 10]

Cf. A174962 (determinants), A386974.

M[j_,k_,n_]:=If[j!=k,j,If[j==k,n+j]]; a[n_]:=Permanent[Table[M[i,j,n],{i,n},{j,n}]];Join[{1}, Array[a,17]]

a(n) = matpermanent(matrix(n, n, j, k, if (j==k, n+j, j))); \\ Michel Marcus, Aug 12 2025

A385570 a(n) is the least positive integer k such that 10^(k/A386725(k)) has at least n correct decimal digits of Pi.

Original entry on oeis.org

1, 1, 81, 85, 87, 785, 1221, 5669, 5669, 193967, 279002, 296009, 893696, 893696, 42601399, 48857271, 98608238, 936331413

Offset: 1

Stefano Spezia, Jul 31 2025

			   n   approximated value of Pi
   -   ------------------------
   1    3.16227766016837933...
   2    3.16227766016837933...
   3    3.14002073072468031...
   4    3.14105848907489924...
   5    3.14154190501421816...
   6    3.14159426168000789...
   7    3.14159239635221012...
   8    3.14159265464843386...
   9    3.14159265464843386...
  10    3.14159265302248860...
  11    3.14159265351804869...
  12    3.14159265358299421...
  13    3.14159265358975264...
  14    3.14159265358975264...
  15    3.14159265358979960...
  16    3.14159265358979359...
  17    3.14159265358979322...
  ...

Index entries for sequences related to the number Pi.

Cf. A000796, A386725.

A386725[n_]:=10^(n/Round[n/Log10[Pi]]); a[n_]:=Module[{k=1},While[RealDigits[A386725[k],10,n][[1]]!=RealDigits[Pi,10,n][[1]], k++]; k]; Array[a,10]

A386823 Triangle read by rows: T(n,k) = numerator((n^2 - k^2)/(n^2 + k^2)), where 0 <= k < n.

Original entry on oeis.org

1, 1, 3, 1, 4, 5, 1, 15, 3, 7, 1, 12, 21, 8, 9, 1, 35, 4, 3, 5, 11, 1, 24, 45, 20, 33, 12, 13, 1, 63, 15, 55, 3, 39, 7, 15, 1, 40, 77, 4, 65, 28, 5, 16, 17, 1, 99, 12, 91, 21, 3, 8, 51, 9, 19, 1, 60, 117, 56, 105, 48, 85, 36, 57, 20, 21, 1, 143, 35, 15, 4, 119, 3, 95, 5, 7, 11, 23

Offset: 1

Stefano Spezia, Aug 04 2025

			The triangle of the fractions begins as:
  1/1;
  1/1,   3/5;
  1/1,   4/5,  5/13;
  1/1, 15/17,   3/5,  7/25;
  1/1, 12/13, 21/29,  8/17,  9/41;
  1/1, 35/37,   4/5,   3/5,  5/13, 11/61;
  1/1, 24/25, 45/53, 20/29, 33/65, 12/37, 13/85;
  ...

Stefano Spezia, First 150 rows of the triangle, flattened

Cf. A000012 (k=0), A000290, A005408, A066830 (k=1), A069011, A094728, A386824 (denominators).

T[n_,k_]:=Numerator[(n^2-k^2)/(n^2+k^2)]; Table[T[n,k],{n,12},{k,0,n-1}]//Flatten

T(n,n-1) = A005804(n-1).

cp's OEIS Frontend

User: Stefano Spezia

A387247 Decimal expansion of (2*log(3) + 7)/8.

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A387235 Decimal expansion of 2*log(2)/3.

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A387131 Decimal expansion of the real part of the complex solutions to log(z) = -1/z on the principal branch of log(z).

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A387132 Decimal expansion of the absolute value of the imaginary part of the complex solutions to log(z) = -1/z on the principal branch of log(z).

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A387101 Decimal expansion of the smallest real solution to e^x = x^3.

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A387102 Decimal expansion of the largest real solution to e^x = x^3.

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A386974 a(n) is the permanent of the symmetric n X n matrix M_n where M_n(j,k) = n for j = k, M_n(j,n) = n-j, M_n(n,k) = n-k, M_n(j,k) = 0 otherwise.

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A386975 a(n) is the permanent of the n X n matrix M_n with M_n(j,k) = j for j <> k, M_n(j,k) = n+j for j = k.

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