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 1-7 of 7 results.

A073247 Squarefree numbers k such that k-1 and k+1 are not squarefree.

Original entry on oeis.org

17, 19, 26, 51, 53, 55, 89, 91, 97, 127, 149, 151, 161, 163, 170, 197, 199, 233, 235, 241, 249, 251, 269, 271, 293, 295, 305, 307, 337, 339, 341, 349, 362, 377, 379, 413, 415, 449, 451, 485, 487, 489, 491, 521, 523, 530, 551, 557, 559, 577, 579, 593, 595
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 22 2002

Keywords

Comments

Probably 11*n < a(n) < 12*n for n > 189. - Charles R Greathouse IV, Nov 05 2017
The asymptotic density of this sequence is 1/zeta(2) - 2 * Product_{p prime} (1 - 2/p^2) + Product_{p prime} (1 - 3/p^2) = A059956 - 2*A065474 + A206256 = 0.088145884881346585838... . - Amiram Eldar, Aug 30 2024

Links

Crossrefs

Cf. A005117, A073249, A073248, A007675.
Cf. A268331, A268332, A268333, A268334 (squarefree numbers isolated by more than 2, 3, etc.).
Cf. A059956, A065474, A206256.

Programs

  • Maple
    sf:= select(numtheory:-issqrfree,[$1..1000]):
map(t -> `if`(sf[t-1]=sf[t]-1 or sf[t+1]=sf[t]+1,NULL,sf[t]), [$2..nops(sf)-1]); # Robert Israel, Feb 01 2016
  • Mathematica
    Reap[For[n = 0, n <= 1000, n++, If[SquareFreeQ[n] && !SquareFreeQ[n-1] && !SquareFreeQ[n+1], Sow[n]]]][[2, 1]] (* Jean-François Alcover, Feb 26 2019 *)
  • PARI
    is(n)=!issquarefree(n-1) && issquarefree(n) && !issquarefree(n+1) \\ Charles R Greathouse IV, Nov 05 2017
  • PARI
    list(lim)=my(v=List(),l1,l2); forfactored(k=9,lim\1+1, if(!issquarefree(k) && !issquarefree(l2) && issquarefree(l1), listput(v,l1[1])); l2=l1; l1=k); Vec(v) \\ Charles R Greathouse IV, Nov 27 2024

A378039 a(1)=3; a(n>1) = n-th first difference of A120327(k) = least nonsquarefree number greater than k.

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Nov 18 2024

Keywords

Comments

The union is {0,1,2,3,4}.

Crossrefs

Positions of 0's are A005117.
Positions of 4's are A007675 - 1, except first term.
Positions of 1's are A068781.
Positions of 2's are A073247 - 1.
Positions of 3's are A073248 - 1, except first term.
First-differences of A120327.
For prime-powers we have A377780, first-differences of A000015.
Restriction is A377784 (first-differences of A377783, union A378040).
The opposite is A378036 (differences A378033), for prime-powers A377782.
The opposite for squarefree is A378085, differences of A070321
For squarefree we have A378087, restriction A378037, differences of A112926.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.
A061398 counts squarefree numbers between primes, zeros A068360.
A061399 counts nonsquarefree numbers between primes, zeros A068361.
Cf. A013928, A053797, A053806, A072284, A224363, A377047, A377049, A378032, A378034, A378082, A378083, A378084.

Programs

  • Mathematica
    Differences[Table[NestWhile[#+1&,n,#>1&&SquareFreeQ[#]&],{n,100}]]

A378085 First differences of A070321 (greatest squarefree number <= n).

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Dec 04 2024

Keywords

Examples

			The greatest squarefree number <= 50 is 47, and the greatest squarefree number <= 51 is 51, so a(51) = 4.

Crossrefs

Ones are A007674.
Zeros are A013929 - 1.
Twos are A280892.
Positions of first appearances are A020755 - 1 (except first term).
First-differences of A070321.
The nonsquarefree restriction is A378034, differences of A378032.
For nonsquarefree numbers we have A378036, differences of A378033.
The opposite restriction to primes is A378037, differences of A112926.
The restriction to primes is A378038, differences of A112925.
The nonsquarefree opposite is A378039, restriction A377784.
The opposite version is A378087.
A005117 lists the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.
Cf. A013928, A020754, A045882, A053797, A053806, A067535, A068781, A072284, A073247, A073248, A120327.

Programs

  • Mathematica
    Differences[Table[NestWhile[#-1&,n,#>1&&!SquareFreeQ[#]&],{n,100}]]

A378087 First-differences of A067535 (least positive integer >= n that is squarefree).

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Nov 20 2024

Keywords

Comments

Does this contain all nonnegative integers? The positions of first appearances begin: 4, 1, 3, 7, 47, 241, 843, 22019, 217069, ...

Crossrefs

Ones are A007674.
Zeros are A013929, complement A005117.
Positions of first appearances are A020754 (except first term) = A045882 - 1.
First-differences of A067535.
Twos are A280892.
For prime-powers we have A377780, differences of A000015.
The nonsquarefree opposite is A378036, differences of A378033.
The restriction to primes + 1 is A378037 (opposite A378038), differences of A112926.
For nonsquarefree numbers we have A378039, see A377783, A377784, A378040.
The opposite is A378085, differences of A070321.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.
A061398 counts squarefree numbers between primes, zeros A068360.
A061399 counts nonsquarefree numbers between primes, zeros A068361.
Cf. A005117, A007675, A068781, A073247, A073248, A378084.
Cf. A013928, A053797, A053806, A072284, A120327, A224363.

Programs

  • Mathematica
    Differences[Table[NestWhile[#+1&,n,#>1&&!SquareFreeQ[#]&],{n,100}]]

A073250 Nonprime squarefree numbers n such that n+1 is also squarefree and nonprime, but not n-1 and n+2.

Original entry on oeis.org

14, 21, 38, 57, 65, 69, 77, 105, 110, 114, 118, 122, 129, 133, 145, 154, 158, 165, 177, 182, 194, 205, 209, 221, 230, 237, 246, 258, 273, 290, 298, 309, 318, 326, 329, 334, 345, 354, 357, 365, 370, 381, 385, 390, 398, 402, 406, 410, 417, 426, 429, 434, 437
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 22 2002

Keywords

Links

Crossrefs

Cf. A000469, A005117, A073248, A073251, A073249.

Programs

  • Mathematica
    tQ[n_]:=!PrimeQ[n+1]&&SquareFreeQ[n+1]&&(PrimeQ[n-1]||!SquareFreeQ[n-1])&&(PrimeQ[n+2]||!SquareFreeQ[n+2])
Select[Select[Complement[Range[500],Prime[Range[PrimePi[500]]]],SquareFreeQ],tQ]  (* Harvey P. Dale, Feb 14 2011 *)
SequencePosition[Table[If[SquareFreeQ[n]&&!PrimeQ[n],1,0],{n,500}],{0,1,1,0}][[;;,1]]+1 (* Harvey P. Dale, Feb 27 2023 *)

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) *)
Showing 1-7 of 7 results.

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