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.

Showing 1-5 of 5 results.

A270344 Least squarefree number that has runs of exactly n nonsquarefree numbers directly preceding and succeeding it.

Original entry on oeis.org

2, 17, 26, 2526, 5876126, 8061827, 8996188226, 5295909711327
Offset: 0

Views

Author

Hartmut F. W. Hoft, Mar 15 2016

Keywords

Comments

The sequence appears to be monotone increasing, but the Mathematica implementation does not assume that.
Only a(0) and a(7) differ from A268330 in the numbers computed so far.

Crossrefs

Cf. A268330.

Programs

  • Mathematica
    (* computes a(n), 1 <= n <= 5 *)
sfRun[n_]:=Module[{i=n}, While[SquareFreeQ[i], i++]; i-n]
nsfRun[n_]:=Module[{i=n}, While[!SquareFreeQ[i], i++]; i-n]
a270344[{low_, high_}]:=Module[{i, next, r, s, list=Table[{},5]}, For[i=low, i<=high, i+=next, r=nsfRun[i]; If[r==0, next=sfRun[i], s=nsfRun[i+r+1]; If[r==s, If[list[[r]]=={}, list[[r]]={i, i+r, i+2r}]; next=2r+2, next=r+1]]]; list]
Map[#[[2]]&, a270344[{0,10000000}]] (* data *)

Extensions

a(7) from Giovanni Resta, Apr 11 2016

A270996 T(i, j) = k is the least squarefree number with a run of exactly i>=0 nonsquarefree numbers immediately preceding k and a run of exactly j>=0 nonsquarefree numbers immediately succeeding k.

Original entry on oeis.org

2, 1, 3, 10, 17, 7, 101, 149, 151, 47, 246, 51, 26, 97, 8474, 1685, 8479, 727, 1861, 241, 843, 22026, 849, 3178, 2526, 10826, 30247, 22019, 217077, 190453, 813251, 55779, 183553, 5045, 580847, 826823
Offset: 0

Views

Author

Hartmut F. W. Hoft, Mar 28 2016

Keywords

Comments

The sequence a(n) = T(i, j) represents the traversal of this matrix by its successive rising antidiagonals.
a(2*i*(i+1)) = A270344(i), for all i >= 0.

Examples

			a(13) = T(1, 3) = 97 since 96, 98, 99 and 100 are nonsquarefree while 95, 97, and 101 are squarefree, and 97 is the smallest number surrounded by the 1,3 pattern.
The matrix T(i, j) with first 8 complete antidiagonals together with some additional elements including the first 7 elements on the diagonal which are A270344(0)..A270344(6):
-------------------------------------------------------------------------
i\j      0       1       2       3        4         5          6        7
-------------------------------------------------------------------------
0:       2       3       7      47     8474       843      22019   826823
1:       1      17     151      97      241     30247     580847   217069
2:      10     149      26    1861    10826      5045     204322 16825126
3:     101      51     727    2526   183553   1944347   28591923 43811049
4:     246    8479    3178   55779  5876126  19375679   67806346
5:    1685     849  813251  450553 29002021   8061827 2082929927
6:   22026  190453  200854 4100277 97447622 245990821 8996188226
7:  217077  826831 7507930 90557979
T(6, 5) = 245990821, T(5, 6) = 2082929927, and all numbers in antidiagonal 11 are larger than 10^8.

Links

Crossrefs

Cf. A005117, A007675, A068088, A073247, A073248, A073251, A228649, A268330-A268334, A270344.

Programs

  • Mathematica
    (* The function computes the least number in the specified interval *)
nsfRun[n_] := Module[{i=n}, While[!SquareFreeQ[i], i++]; i-n]
a270996[{low_, high_},{widthL_, widthR_}] := Module[{i=low, r, s, first=0}, While[i<=high, r=nsfRun[i]; If[r != widthL, i+=r+1, s=nsfRun[i+r+1]; If[s != widthR, If[s != widthL, i+=r+s+2, i+=r+1], first=i+r; i=high+1]]]; first]
a270996[{0, 5000},{2, 3}] (* computes a(18) = T(2, 3) *)

A271145 a(n) = k is the least number at which an isolated alternating run of nonsquarefree/squarefree (nsf/sf) numbers of size n starts.

Original entry on oeis.org

2, 14, 482, 6346
Offset: 0

Views

Author

Hartmut F. W. Hoft, Mar 31 2016

Keywords

Comments

A contiguous sequence of numbers satisfying the pattern sf sf nsf sf ... nsf sf nsf sf sf with k+1 nsf numbers alternating with k sf numbers that are bounded by a pair of sf numbers at both ends is called an isolated alternating nsf/sf run of size k. The left sf bounding number is the start of the run.
Any such run must start at an even number i and have an even size j, since for i odd i+3 is nsf, and for i even and j odd i+2*j+4 is nsf.
For all n>=0, a(n)+2 is divisible by 4.
a(4) > 5*10^9

Examples

			a(0) = 2 since 2, 3, 5 and 6 are sf while 4 is nsf.
a(2) = 482 since in the interval 482...494 the nsf/sf pattern is sf sf nsf sf nsf sf nsf sf nsf sf nsf sf sf and it is the first occurrence of that 13-number run.

Crossrefs

Cf. A005117, A073247, A073248, A268330, A270344.

Programs

  • Mathematica
    nsfRun[n_] := Module[{i=n}, While[!SquareFreeQ[i], i++]; i-n]
sfRun[n_] := Module[{i=n}, While[SquareFreeQ[i], i++]; i-n]
sfBlockSearch[i_] := Module[{searching=True, j=i, r, s}, While[searching, r=nsfRun[j]; s=sfRun[j+r]; If[s<2, j+=r+s, searching=False]]; j+r+s]
nsfsfPairQ[i_] := nsfRun[i]==1 && sfRun[i+1]==1
nsfsfEndQ[i_] := nsfRun[i]==1 && sfRun[i+1]>1
nsfsfRun[i_] := Module[{searching=True, count, j=i, s, e}, j=sfBlockSearch[j]; While[searching, count=0; s=j; While[nsfsfPairQ[j], count++; j+=2]; e=j; If[count==0 || !nsfsfEndQ[j], j=sfBlockSearch[j], searching=False]]; {s, e, count}]
a271145[{low_, high_}, b_] := Module[{i=low, k, k3, list=Table[{}, b]}, While[i<=high, k=nsfsfRun[i]; k3=Last[k]/2; If[list[[k3]]=={}, list[[k3]]=k[[1]]-2]; i=k[[2]]]; list]
a271145[{0, 10000}, 3] (* computes a(1), a(2), a(3) *)

A271041 a(n) = k is the least squarefree number with a run of zero nonsquarefree numbers immediately preceding k and a run of exactly n>=0 nonsquarefree numbers immediately succeeding k.

Original entry on oeis.org

2, 3, 7, 47, 8474, 843, 22019, 826823, 1092746, 33908367, 262315466
Offset: 0

Views

Author

Hartmut F. W. Hoft, Mar 29 2016

Keywords

Comments

This is the top row (indexed by zero) of the doubly infinite matrix T(i,j) defined in A270996: a(n) = A270996((n+1)*(n+2)/2-1).
a(11) = T(0,11) > 3*10^9.

Examples

			a(3) = 47 since 46, 47 and 51 are squarefree while 48, 49 and 50 constitute the first nonsquarefree triple after a pair of squarefree numbers.

Crossrefs

Cf. A005117, A268330, A270344, A270996.

Programs

  • Mathematica
    (* We use function a270996[] from A270996 *)
Map[a270996[{0, 1000000},{0, #}]&,Range[0,7]]
(* computes a(0)...a(7) *)

A271042 a(n) = k is the least squarefree number with a run of n >= 0 nonsquarefree numbers immediately preceding k and a run of zero nonsquarefree numbers immediately succeeding k.

Original entry on oeis.org

2, 1, 10, 101, 246, 1685, 22026, 217077, 7216626, 8870033, 262315477, 221167433, 105779897634, 82462576233
Offset: 0

Views

Author

Hartmut F. W. Hoft, Mar 29 2016

Keywords

Comments

This is the leftmost column (indexed by zero) of the doubly infinite matrix T(i,j) defined in A270996: a(n) = A270996(n*(n+1)/2).

Examples

			a(3) = 101 since 97, 101 and 102 are squarefree while 98, 99 and 100 constitute the first nonsquarefree triple followed by a pair of squarefree numbers.

Crossrefs

Cf. A005117, A268330, A270344, A270996.

Programs

  • Mathematica
    (* We use function a270996[] from A270996 *)
Map[a270996[{0, 1000000},{#, 0}]&,Range[0,7]]
(* computes a(0)...a(7) *)

Extensions

a(12)-a(13) from Kevin P. Thompson, Apr 19 2022
Showing 1-5 of 5 results.

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

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