cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: Michael Kleber

Michael Kleber's wiki page.

Michael Kleber has authored 40 sequences. Here are the ten most recent ones:

A325409 Number of (2n+1) X (2n+1) black-and-white mirror-symmetric grids that are legal for crossword puzzles and have no all-black edges.

Original entry on oeis.org

1, 2, 20, 269, 8116, 519400, 67566361, 17518353991, 9334778743018, 10240638005472594, 23244895727823155064
Offset: 1

Views

Author

Michael Kleber, Apr 23 2019

Keywords

Comments

Rules: Every white square must be "checked" (in both a horizontal and vertical word), words must be at least 3 letters long, the whole grid must be symmetric under both left-right and top-bottom mirror symmetry, and the set of white squares must be connected.

Crossrefs

Cf. A323838, A323839, A325408.
There are many other OEIS sequences related to crossword puzzles: search for "crossword puzzle" (without the quotes).

Extensions

a(11) added by Akshay Ravikumar, Jun 20 2019

A325408 Number of (2n+1) X (2n+1) black-and-white mirror-symmetric grids that are legal for crossword puzzles.

Original entry on oeis.org

1, 5, 35, 436, 11348, 650798, 78816939, 19500572516, 10050221059942, 10777659213636406, 24081651896803403442
Offset: 1

Views

Author

Michael Kleber, Apr 23 2019

Keywords

Comments

Rules: Every white square must be "checked" (in both a horizontal and vertical word), words must be at least 3 letters long, the whole grid must be symmetric under both left-right and top-bottom mirror symmetry, and the set of white squares must be connected.

Crossrefs

Cf. A323838, A323839, A325409.
There are many other OEIS sequences related to crossword puzzles: search for "crossword puzzle" (without the quotes).

Extensions

a(11) added by Akshay Ravikumar, Jun 20 2019

A253269 Weakly Twin Primes in base 10: Can only reach one other prime by single-decimal-digit changes.

Original entry on oeis.org

89391959, 89591959, 519512471, 519512473, 531324041, 561324041, 699023791, 699023891, 874481011, 874487011, 1862537503, 2232483271, 2232483871, 2608559351, 3127181789, 3157181789, 3928401949, 3928401989, 4070171669, 4070171969, 5225628323, 5309756339, 5525628323
Offset: 1

Views

Author

Michael Kleber, May 01 2015

Keywords

Comments

Each pair of twins here form a size-two connected component in the graph considered in A158576.
A naive heuristic argument based on the density of primes claims that this sequence should be infinite, and in fact that a positive proportion of all primes should have this property. A prime p has 9*log_10(p) neighbors, each prime with "probability" 1/log(p), and with all the other 2*9*log_10(p) neighbors being composite with "probability" (1-1/log(p))^(2*9*log_10(p)). For a large prime p, this goes to the limit 9/(exp(18/log(10))*log(10)), or about 0.16%. The fact that base-10 primes need to end with digit 1/3/7/9 will change the value of this probability, but won't change the fact that it is nonzero.
This is analogous to a theorem about weakly prime numbers; see the Terence Tao paper referenced in A050249.

Links

Crossrefs

Cf. A158576, A050249, A158124.

Programs

  • Mathematica
    NeighborsAndSelf[n_] := Flatten[MapIndexed[Table[ n + (i - #)*10^(#2[[1]] - 1), {i, 0, 9}] &, Reverse[IntegerDigits[n, 10]]]]
PrimeNeighbors[n_] := Complement[Select[NeighborsAndSelf[n],PrimeQ],{n}]
WeaklyTwinPrime[p_] := (Length[#] == 1 && PrimeNeighbors[#[[1]]] == {p}) &[PrimeNeighbors[p]]
For[k = 0, k <= PrimePi[10^10], k++, If[WeaklyTwinPrime[Prime[k]], Print[Prime[k]]]]

A152549 Decimal expansion of log_3(18).

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Oct 28 2009, based on a posting by Michael Kleber to the Math Fun Mailing List

Keywords

Comments

Hausdorff dimension of Jeannine Mosely's origami model of Menger's sponge.

Examples

			2.6309297535714574370995271143427608542995856401318804278706...

Links

Programs

Formula

Equals 2+A102525. - R. J. Mathar, Oct 02 2023

A135707 Consider a domino formed from two adjacent 1 X 1 squares. This is the decimal expansion of the average distance between a random point in the left square and a random point in the right square.

Original entry on oeis.org

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

Views

Author

Richard C. Schroeppel and Michael Kleber, Mar 05 2008

Keywords

Examples

			1.0881382498612540260133800398822187615937033186594483235371...

Programs

  • Mathematica
    d[x1_,y1_,x2_,y2_] := Sqrt[ (x1-x2)^2+(y1-y2)^2 ]
Integrate[ d[x1,y1,x2,y2], {x1,-1,0},{x2,0,1},{y1,0,1},{y2,0,1} ]

Formula

(116 - 8 Sqrt[2] - 20 Sqrt[5] + 140 ArcCsch[2] - 40 ArcSinh[1] + 80 ArcSinh[2] + Log[32] + 10 Log[-1 + Sqrt[5]] - 15 Log[123 + 55 Sqrt[5]]) / 120

A132468 Longest gap between numbers relatively prime to n.

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 3, 1, 3, 2, 1, 1, 3, 1, 3, 2, 3, 1, 3, 1, 3, 1, 3, 1, 5, 1, 1, 2, 3, 2, 3, 1, 3, 2, 3, 1, 5, 1, 3, 2, 3, 1, 3, 1, 3, 2, 3, 1, 3, 2, 3, 2, 3, 1, 5, 1, 3, 2, 1, 2, 5, 1, 3, 2, 5, 1, 3, 1, 3, 2, 3, 2, 5, 1, 3, 1, 3, 1, 5, 2, 3, 2, 3, 1, 5, 2, 3, 2, 3, 2, 3, 1, 3, 2, 3, 1, 5, 1, 3, 4
Offset: 1

Views

Author

Michael Kleber, Nov 16 2007

Keywords

Comments

Here "gap" does not include the endpoints.
a(n) is given by the maximum length of a run of numbers satisfying one congruence modulo each of n's distinct prime factors. It follows that if m is the number of distinct prime factors of n and each of n's prime factors is greater than m then a(n) = m. - Thomas Anton, Dec 30 2018

Examples

			E.g. n=3: the longest gap in 1, 2, 4, 5, 7, ... is 1, between 2 and 4, so a(3) = 1.

Links

Crossrefs

Equals A048669(n) - 1.
See also A048670, A049298, A070791, A070194.

Programs

  • Maple
    a:=[];
for n from 1 to 120 do
s:=[seq(j,j=1..4*n)];
rec:=0;
   for st from 1 to n do
   len:=0;
    for i from 1 to n while gcd(s[st+i-1],n)>1 do len:=len+1; od:
    if len>rec then rec:=len; fi;
   od:
a:=[op(a),rec];
od:
a; # N. J. A. Sloane, Apr 18 2017
  • Mathematica
    a[ n_ ] := (Max[ Drop[ #,1 ]-Drop[ #,-1 ] ]-1&)[ Select[ Range[ n+1 ],GCD[ #,n ]==1& ] ]
Do[Print[n, " ", a[n]],{n,20000}]

Formula

a(n) = 1 at every prime power.

Extensions

Incorrect formula removed by Thomas Anton, Dec 30 2018

A131746 Second French version of A131744.

Original entry on oeis.org

9, 16, 15, 13, 14, 4, 17, 21, 12, 4, 12, 5, 12, 21, 15, 2, 13, 4, 17, 21, 12, 4, 20, 19, 5, 3, 8, 21, 12, 4, 20, 19, 2, 13, 1, 5, 15, 5, 14, 11, 4, 2, 13, 5, 7, 13, 15, 15, 1, 7, 10, 11, 6, 5, 21, 12, 4, 20, 19, 2, 13, 1, 11, 6, 5, 21, 2, 6, 5, 3, 13, 11, 6, 5, 21, 17, 13, 5, 7
Offset: 1

Views

Author

Michael Kleber, Sep 20 2007

Keywords

Comments

Diacritics are ignored when computing the rank of a letter.
The numbers 18 and 22-25 do not occur in the sequence. - David Applegate, Sep 24 2007

Examples

			The sequence begins Neuf, seize, quinze, treize, quatorze, ...

Crossrefs

Cf. A131744, A131745, A130316, A133816, A133817.

Extensions

More terms from David Applegate, Sep 24 2007

A131745 First French version of A131744.

Original entry on oeis.org

4, 20, 19, 2, 13, 17, 13, 5, 7, 13, 16, 5, 15, 10, 9, 16, 15, 2, 1, 16, 3, 4, 2, 13, 4, 17, 21, 1, 5, 15, 5, 14, 11, 4, 0, 2, 13, 4, 17, 21, 2, 6, 5, 3, 2, 14, 11, 4, 0, 2, 13, 4, 17, 21, 14, 14, 4, 17, 21, 2, 6, 5, 3, 0, 4, 12, 5, 12, 21, 1, 5, 15, 10, 9, 16, 15, 13, 14, 4, 17
Offset: 1

Views

Author

Michael Kleber, Sep 20 2007

Keywords

Comments

Diacritics are ignored when computing the rank of a letter.
Spellings used for French numbers: zero, un, deux, trois, quatre, cinq, six, sept, huit, neuf, dix, onze, douze, treize, quatorze, quinze, seize, dix-sept, dix-huit, dix-neuf, vingt, vingt et un, vingt-deux, vingt-trois, vingt-quatre, vingt-cinq.
The numbers 18 and 22-25 do not occur in the sequence. - David Applegate, Sep 24 2007
There appears to be no analog in Spanish. (The requirement is that the difference between the ranks of the first two letters in the word for the number n should equal n.) - David Applegate, Sep 24 2007. There exists such a sequence (namely, 12, 13, 2, 16, 2, 14, 2, 2, 1...) if (as is standard) letter 'ñ' is granted an alphabet position, whereas digraphs such as 'll' are not. - Álvar Ibeas, Sep 18 2020

Examples

			The sequence begins Quatre, vingt, dix-neuf, deux, treize, ...

Crossrefs

Cf. A131744, A131746, A130316, A133816, A133817.

Extensions

More terms from David Applegate, Sep 24 2007

A096244 Number of n-digit base-11 deletable primes.

Original entry on oeis.org

4, 16, 73, 288, 1117, 4472, 18120, 74643, 315174, 1348936
Offset: 1

Views

Author

Michael Kleber, Feb 28 2003

Keywords

Comments

A prime p is a base-b deletable prime if when written in base b it has the property that removing some digit leaves either the empty string or another deletable prime. "Digit" means digit in base b.
Deleting a digit cannot leave any leading zeros in the new string. For example, deleting the 2 in 2003 to obtain 003 is not allowed.

Crossrefs

Cf. A080608, A080603, A096235-A096246.

Programs

  • Mathematica
    b = 11; a = {4}; d = {2, 3, 5, 7};
For[n = 2, n <= 5, n++,
  p = Select[Range[b^(n - 1), b^n - 1], PrimeQ[#] &];
  ct = 0;
  For[i = 1, i <= Length[p], i++,
   c = IntegerDigits[p[[i]], b];
   For[j = 1, j <= n, j++,
    t = Delete[c, j];
    If[t[[1]] == 0, Continue[]];
    If[MemberQ[d, FromDigits[t, b]], AppendTo[d, p[[i]]]; ct++;
     Break[]]]];
  AppendTo[a, ct]];
a (* Robert Price, Nov 13 2018 *)

Extensions

5 more terms from Ryan Propper, Jul 19 2005

A096242 Number of n-digit base-9 deletable primes.

Original entry on oeis.org

4, 14, 58, 221, 911, 3638, 14687, 61435, 262189, 1140171
Offset: 1

Views

Author

Michael Kleber, Feb 28 2003

Keywords

Comments

A prime p is a base-b deletable prime if when written in base b it has the property that removing some digit leaves either the empty string or another deletable prime. "Digit" means digit in base b.
Deleting a digit cannot leave any leading zeros in the new string. For example, deleting the 2 in 2003 to obtain 003 is not allowed.

Crossrefs

Cf. A080608, A080603, A096235-A096246.

Programs

  • Mathematica
    b = 9; a = {4}; d = {2, 3, 5, 7};
For[n = 2, n <= 5, n++,
  p = Select[Range[b^(n - 1), b^n - 1], PrimeQ[#] &];
  ct = 0;
  For[i = 1, i <= Length[p], i++,
   c = IntegerDigits[p[[i]], b];
   For[j = 1, j <= n, j++,
    t = Delete[c, j];
    If[t[[1]] == 0, Continue[]];
    If[MemberQ[d, FromDigits[t, b]], AppendTo[d, p[[i]]]; ct++;
     Break[]]]];
  AppendTo[a, ct]];
a (* Robert Price, Nov 13 2018 *)

Extensions

5 more terms from Ryan Propper, Jul 19 2005

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.