Jodi Spitz - cp's OEIS frontend

A362810 Define G(n, k) to be the n-th derivative of Gamma(x) at k. a(n)=floor(min(G(2n, x))), where min(f) is the local minimum of f in [0,oo).

0, 0, 1, 6, 30, 173, 1138, 8386, 67951, 596745, 5618916, 56249658, 594648335, 6602123630, 76631632344, 926329705808, 11623455427764, 150970962492188, 2024773236657401, 27980260971851306, 397645587914766071, 5801999753304428181, 86784442260270596447, 1328924296505789704631, 20807559990139289975657, 332753116291423840918784

Offset: 0

Jodi Spitz, May 04 2023

nonn

Appears to grow factorially (superexponentially).

Conjecture: limit_{n->oo} log(a(n)) / log(n!) < 1. - Vaclav Kotesovec, Nov 17 2023

			a(5) = 173 since the local minimum in [0,oo) of the 10th derivative of Gamma(x) is 173.195...

Cf. A030171.

Join[{0}, Floor[Table[d = Simplify[D[Gamma[x], {x, 2 n}]]; d /. FindRoot[D[d, x] == 0, {x, n/2}, WorkingPrecision -> 50], {n, 1, 10}]]] (* Vaclav Kotesovec, Nov 17 2023 *)

a(7)-a(25) from Vaclav Kotesovec, Nov 18 2023

A362094 Number of connected supports with n standard pieces for standard puzzles of the shape 2 X k, up to support-reduction. (See comments and reference for precise definition.)

Original entry on oeis.org

6, 37, 259, 1391, 5460

Offset: 1

Jodi Spitz, Apr 08 2023

A piece is a 2 X 2 matrix of distinct numbers, each called a label. A standard piece is a 2 X 2 matrix containing, in some order, the numbers {1,2,3,4} once each. A piece p_1 can be reduced to a standard piece p_2 if p_2 preserves the label order of p_1. For example,
   6--17                              2--4
   |  | reduces to the standard piece |  |.
   9--5                               3--1
A standard puzzle of the shape 2 X k is a 2 X k matrix containing, in some order, {1,2,...,2k}. A support P for a standard puzzle Q of the shape 2 X k is a finite set of standard pieces {p_1,p_2,...} such that for any 2 X 2 submatrix T of Q, there exists a p_x in P such that T is equivalent to p_x under reduction.
A support P is connected if for any two pieces p_1, p_2 in P, there exists a standard puzzle containing p_1 and p_2 in its support. Two supports P, P' are equivalent under support-reduction if P' can be reached from P by: 1) exchanging the left and right columns of every piece in P, 2) exchanging the top and bottom row of every piece in P, and/or 3) replacing each label c of every piece in P with (5-c).
Note: Han (see Links) simply calls support-reduction "reduction." It has been called "support-reduction" here to distinguish it from the reduction of pieces into standard pieces.
For further definitions and clarification, see Han reference.

			a(1) = 6. There exist 4! = 24 standard pieces and so 24 unique supports P with 1 standard piece. Of these supports, there is at most a set of a(1) = 6 supports which cannot be support-reduced to each other, such as:
   4--3     3--4     4--2     2--4     3--2         2--3
  {|  |} , {|  |} , {|  |} , {|  |} , {|  |} , and {|  |} .
   1--2     1--2     1--3     1--3     1--4         1--4
We know these supports are connected because for any of support from this set P and any 2 standard pieces p_1, p_2 in P, there exists a standard puzzle with p_1 and p_2 in its support. (This is obvious since each support has only 1 piece.)

Guo-Niu Han, Enumeration of Standard Puzzles, University of Strasbourg, May 2011, page 5.

Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.

Cf. A196265.

A362695 Decimal expansion of (3 - sqrt(3))/4.

Original entry on oeis.org

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

Offset: 0

Jodi Spitz, Apr 29 2023

Consider the optimal packing of 2 identical spheres in a cube of edge length 1. The radius of both spheres is (3 - sqrt(3))/4 = 0.3169872....

			0.3169872981077806766181384146235...

Packomania, The best known packings of equal spheres in a cube.
Stack Exchange, Two Sphere [sic] inside a cube.

Cf. A334843 (their diameter).

RealDigits[(3 - Sqrt[3])/4, 10, 120][[1]] (* Amiram Eldar, Jun 25 2023 *)

A361838 a(n) is the number of 2s in the binary hereditary representation of 2n.

Original entry on oeis.org

1, 2, 3, 2, 3, 4, 5, 3, 4, 5, 6, 5, 6, 7, 8, 3, 4, 5, 6, 5, 6, 7, 8, 6, 7, 8, 9, 8, 9, 10, 11, 4, 5, 6, 7, 6, 7, 8, 9, 7, 8, 9, 10, 9, 10, 11, 12, 7, 8, 9, 10, 9, 10, 11, 12, 10, 11, 12, 13, 12, 13, 14, 15, 4, 5, 6, 7, 6, 7, 8, 9, 7, 8, 9, 10, 9, 10, 11, 12, 7

Offset: 1

Jodi Spitz, Mar 26 2023

See comments on A266201 for the definition of hereditary representation.

			A table of n, the binary hereditary representation of 2n, and the number of 2s in the representation:
 n | hereditary rep. of 2n   | number of 2s
---+-------------------------+--------------
 1 | 2                       |      1
 2 | 2^2                     |      2
 3 | 2^2+2                   |      3
 4 | 2^(2+1)                 |      2
 5 | 2^(2+1)+2               |      3
 6 | 2^(2+1)+2^2             |      4
 7 | 2^(2+1)+2^2+2           |      5
 8 | 2^2^2                   |      3
 9 | 2^2^2+2                 |      4
10 | 2^2^2+2^2               |      5
11 | 2^2^2+2^2+2             |      6
12 | 2^2^2+2^(2+1)           |      5
13 | 2^2^2+2^(2+1)+2         |      6
14 | 2^2^2+2^(2+1)+2^2       |      7
15 | 2^2^2+2^(2+1)+2^2+2     |      8
16 | 2^(2^2+1)               |      3
17 | 2^(2^2+1)+2             |      4
18 | 2^(2^2+1)+2^2           |      5
19 | 2^(2^2+1)+2^2+2         |      6
20 | 2^(2^2+1)+2^(2+1)       |      5
21 | 2^(2^2+1)+2^(2+1)+2     |      6
22 | 2^(2^2+1)+2^(2+1)+2^2   |      7
23 | 2^(2^2+1)+2^(2+1)+2^2+2 |      8
24 | 2^(2^2+1)+2^2^2         |      6
25 | 2^(2^2+1)+2^2^2+2       |      7
26 | 2^(2^2+1)+2^2^2+2^2     |      8
27 | 2^(2^2+1)+2^2^2+2^2+2   |      9
28 | 2^(2^2+1)+2^2^2+2^(2+1) |      8

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A005245, A025280.

a(n)=if(n==0, 0, sum(k=0, logint(n,2), if(bittest(n,k), 1 + a((k+1)\2)))) \\ Andrew Howroyd, Apr 07 2023

A361747 Lexicographically earliest sequence of distinct positive integers such that a(n) and a(n-1) share at least one identical trit at the same position in their balanced ternary representations.

Original entry on oeis.org

1, 4, 2, 3, 6, 5, 7, 8, 9, 10, 11, 12, 13, 16, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 43, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67

Offset: 1

Jodi Spitz, Mar 22 2023

Conjecture: Suppose n is the x-th fixed point such that n-1 is not a fixed point. Then for all k such that n < k < (3^(x + 2) + 1)/2, k is also a fixed point.

			Table of initial terms (T is a digit of the value -1):
  n | a(n) |  BAL
 ---+------+-------
  1 |    1 |    1
  2 |    4 |   11
  3 |    2 |   1T
  4 |    3 |   10
  5 |    6 |  1T0
  6 |    5 |  1TT
  7 |    7 |  1T1
  8 |    8 |  10T
  9 |    9 |  100
 10 |   10 |  101
 11 |   11 |  11T
 12 |   12 |  110
 13 |   13 |  111
 14 |   16 | 1TT1
 15 |   14 | 1TTT
 16 |   15 | 1TT0
 17 |   17 | 1T0T
 18 |   18 | 1T00
 19 |   19 | 1T01
 20 |   20 | 1T1T

A361445 Sums of consecutive terms of A361444.

Original entry on oeis.org

3, 5, 7, 11, 13, 11, 13, 17, 31, 163, 229, 199, 313, 257, 277, 233, 223, 311, 331, 353, 373, 353, 167, 197, 373, 433, 463, 349, 359, 433, 443, 463, 433, 443, 787, 919, 727, 757, 787, 797, 397, 347, 727, 457, 467, 757, 727, 787, 857, 1009, 1019, 857

Offset: 1

Jodi Spitz, Mar 12 2023

Cf. A086527, A361444.

A361377 Squares visited by a knight moving on a spirally numbered board always to the lowest unvisited coprime square.

Original entry on oeis.org

1, 10, 3, 8, 5, 2, 7, 4, 9, 22, 19, 16, 33, 58, 13, 28, 25, 46, 21, 40, 17, 6, 23, 20, 39, 70, 43, 76, 47, 26, 11, 14, 29, 32, 15, 62, 37, 18, 35, 38, 63, 34, 59, 30, 53, 12, 31, 54, 85, 124, 51, 80, 83, 52, 49, 24, 77, 48, 119, 50, 27, 86, 55, 128, 89, 92

Offset: 1

Jodi Spitz, Mar 09 2023

Many of these sequences (see cross-references) are finite. I've worked this out by hand, but I suspect this sequence is also finite.

The sequence is finite with 156 terms. - Rémy Sigrist, Mar 12 2023

			The spiral board begins:
   .---.---.--33--32--31
                       |
  17--16--15--14--13  30
   |               |   |
  18   5---4---3  12  29
   |   |       |   |   |
  19   6   1---2  11  28
   |   |           |   |
  20   7---8---9--10  27
   |                   |
  21--22--23--24--25--26
a(9) = 9 and a(10) = 22. For a knight on square 9, the smallest unused square which is both coprime to and a knight's move away from 9 is 22.

Rémy Sigrist, Table of n, a(n) for n = 1..156
Rémy Sigrist, PARI program

Cf. A316667, A326922, A328929, A328928.

PARI
```
See Links section.
```

Data corrected by Rémy Sigrist, Mar 12 2023

A361444 Lexicographically earliest sequence of distinct positive base-10 palindromes such that a(n) + a(n+1) is prime.

Original entry on oeis.org

1, 2, 3, 4, 7, 6, 5, 8, 9, 22, 141, 88, 111, 202, 55, 222, 11, 212, 99, 232, 121, 252, 101, 66, 131, 242, 191, 272, 77, 282, 151, 292, 171, 262, 181, 606, 313, 414, 343, 444, 353, 44, 303, 424, 33, 434, 323, 404, 383, 474, 535, 484, 373, 454, 333, 464, 363

Offset: 1

Jodi Spitz, Mar 12 2023

			a(10) = 22, the smallest unused positive palindrome which can be added to a(9) = 9 to get a prime; 9 + 22 = 31.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..10^4, with a color function showing number of decimal digits of a(n) where red = 1, orange = 2, ..., magenta = 8.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..10^4, with a color function showing a(n) mod 10, where red = 1, orange = 2, ..., magenta = 9.

Cf. A002113, A082979.

nn = 60; kk = 5*10^4; c[_] = False; a[1] = j = 1; c[1] = True; u = 2;
MapIndexed[Set[s[First[#2]], #1] &, Select[Range[kk], PalindromeQ]];
Do[k = u; While[Or[c[k], CompositeQ[s[k] + j]], k++];
 Set[{a[n], c[k], j}, {k, True, s[k]}];
 If[k == u, While[c[u], u++]], {n, 2, nn}];
Array[s @* a, nn] (* Michael De Vlieger, Mar 18 2023 *)

nextpal(k) = my(d=digits(k)); while (d!=Vecrev(d), k++; d = digits(k)); k;
lista(nn) = my(va = vector(nn)); va[1] = 1; for (n=2, nn, my(k=1); while(!isprime(va[n-1]+k) || #select(x->(x==k), va), k=nextpal(k+1)); va[n] = k;); va; \\ Michel Marcus, Mar 19 2023

from sympy import isprime
from itertools import count, islice, product
def pals(): # generator of palindromes
    digits = "0123456789"
    for d in count(1):
        for p in product(digits, repeat=d//2):
            if d > 1 and p[0] == "0": continue
            left = "".join(p); right = left[::-1]
            for mid in [[""], digits][d%2]:
                yield int(left + mid + right)
def agen(): # generator of terms of sequence
    pg, passed = pals(), []
    an = next(p for p in pg if p > 0) # start at 1
    while True:
        yield an
        for p in passed:
            if isprime(an+p):
                passed.remove(p)
                break
        else:
             while not isprime(an + (p:=next(pg))):
                 passed.append(p)
        an = p
print(list(islice(agen(), 57))) # Michael S. Branicky, Mar 12 2023

a(25) and beyond from Michael S. Branicky, Mar 12 2023

A361299 Counterclockwise spiral constructed of distinct terms such that any two terms a knight's move apart are coprime; always choose the smallest possible positive term.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 8, 11, 10, 13, 6, 15, 12, 17, 14, 19, 16, 23, 18, 25, 20, 29, 22, 21, 24, 31, 26, 37, 28, 35, 32, 41, 34, 43, 27, 33, 36, 47, 44, 39, 38, 49, 40, 53, 46, 59, 30, 51, 50, 61, 55, 67, 58, 71, 52, 45, 56, 73, 62, 65, 64, 79, 42, 77, 48, 83

Offset: 1

Jodi Spitz, Mar 08 2023

nonn

			The spiral begins:
  33--27--43--34--41--32--35
   |                       |
  36  19--14--17--12--15  28
   |   |               |   |
  47  16   5---4---3   6  37
   |   |   |       |   |   |
  44  23   7   1---2  13  26
   |   |   |           |   |
  39  18   9---8--11--10  31
   |   |                   |
  38  25--20--29--22--21--24
   |
  49--40--53--46---.---.---.

Jodi Spitz, Table of n, a(n) for n = 1..4002
Rémy Sigrist, PARI program

Cf. A308884.

PARI
```
See Links section.
```

More terms from Rémy Sigrist, Mar 09 2023

A359752 Lexicographically earliest array of distinct positive integers read by antidiagonals such that integers in cells which are a knight's move apart are coprime.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 6, 8, 9, 11, 13, 17, 19, 23, 10, 12, 15, 21, 27, 16, 14, 22, 25, 29, 31, 37, 20, 33, 18, 24, 35, 26, 39, 41, 43, 47, 49, 53, 59, 61, 32, 55, 67, 45, 28, 30, 51, 57, 63, 34, 36, 71, 73, 38, 44, 40, 65, 79, 83, 89, 85, 77, 91, 69, 42, 75

Offset: 1

Jodi Spitz, Mar 07 2023

			The array begins:
   1  2  4  6 13 12 22 18 ...
   3  5  8 17 15 25 24 ...
   7  9 19 21 29 35 ...
  11 23 27 31 26 ...
  10 16 37 39 ...
  14 20 41 ...
  33 43 ...
  47 ...

Jodi Spitz, Table of n, a(n) for n = 1..5499
Rémy Sigrist, Colored representation of the first 500 antidiagonals (plus-shaped patterns correspond to even values)
Rémy Sigrist, PARI program

Cf. A097883.

PARI
```
See Links section.
```

More terms from Rémy Sigrist, Mar 09 2023

More terms from Jodi Spitz, Mar 10 2023

cp's OEIS Frontend

User: Jodi Spitz

A362810 Define G(n, k) to be the n-th derivative of Gamma(x) at k. a(n)=floor(min(G(2n, x))), where min(f) is the local minimum of f in [0,oo).

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A362094 Number of connected supports with n standard pieces for standard puzzles of the shape 2 X k, up to support-reduction. (See comments and reference for precise definition.)

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A362695 Decimal expansion of (3 - sqrt(3))/4.

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Crossrefs

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Mathematica

A361838 a(n) is the number of 2s in the binary hereditary representation of 2n.

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Comments

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Crossrefs

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PARI

A361747 Lexicographically earliest sequence of distinct positive integers such that a(n) and a(n-1) share at least one identical trit at the same position in their balanced ternary representations.

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Author

Keywords

Comments

Examples

A361445 Sums of consecutive terms of A361444.

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Keywords

Crossrefs

A361377 Squares visited by a knight moving on a spirally numbered board always to the lowest unvisited coprime square.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions

A361444 Lexicographically earliest sequence of distinct positive base-10 palindromes such that a(n) + a(n+1) is prime.

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Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Extensions

A361299 Counterclockwise spiral constructed of distinct terms such that any two terms a knight's move apart are coprime; always choose the smallest possible positive term.

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Author

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Examples