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 1-4 of 4 results.

A112925 Largest squarefree integer < the n-th prime.

Original entry on oeis.org

1, 2, 3, 6, 10, 11, 15, 17, 22, 26, 30, 35, 39, 42, 46, 51, 58, 59, 66, 70, 71, 78, 82, 87, 95, 97, 102, 106, 107, 111, 123, 130, 134, 138, 146, 149, 155, 161, 166, 170, 178, 179, 190, 191, 195, 197, 210, 222, 226, 227, 231, 238, 239, 249, 255, 262, 267, 269, 274, 278
Offset: 1

Views

Author

Leroy Quet, Oct 06 2005

Keywords

Examples

			6 is the largest squarefree less than the 4th prime, 7. So a(4) = 6.

Links

Crossrefs

For prime powers instead of squarefree numbers we have A065514, opposite A345531.
Restriction of A070321 (differences A378085) to the primes; see A378619.
The opposite is A112926, differences A378037.
Subtracting each term from prime(n) gives A240473, opposite A240474.
For nonsquarefree numbers we have A378033, differences A378036, see A378034, A378032.
For perfect powers we have A378035.
First differences are A378038.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers, differences A076259.
A013928 counts squarefree numbers up to n - 1.
A013929 lists the nonsquarefree numbers, differences A078147.
A061398 counts squarefree numbers between primes, zeros A068360.
A061399 counts nonsquarefree numbers between primes, zeros A068361.
A112929 counts squarefree numbers up to prime(n).
Cf. A061400, A112928, A112930.
Cf. A007674, A007947, A045882, A053797, A053806, A067535, A072284, A073247, A081217, A175216, A280892, A377783.

Programs

  • Maple
    with(numtheory): a:=proc(n) local p,B,j: p:=ithprime(n): B:={}: for j from 1 to p-1 do if abs(mobius(j))>0 then B:=B union {j} else B:=B fi od: B[nops(B)] end: seq(a(m),m=1..75); # Emeric Deutsch, Oct 14 2005
  • Mathematica
    With[{k = 120}, Table[SelectFirst[Range[Prime@ n - 1, Prime@ n - Min[Prime@ n - 1, k], -1], SquareFreeQ], {n, 60}]] (* Michael De Vlieger, Aug 16 2017 *)
  • PARI
    a(n,p=prime(n))=while(!issquarefree(p--),); p \\ Charles R Greathouse IV, Aug 16 2017

Formula

a(n) = prime(n) - A240473(n). - Gus Wiseman, Jan 10 2025

Extensions

More terms from Emeric Deutsch, Oct 14 2005

A112926 Smallest squarefree integer > the n-th prime.

Original entry on oeis.org

3, 5, 6, 10, 13, 14, 19, 21, 26, 30, 33, 38, 42, 46, 51, 55, 61, 62, 69, 73, 74, 82, 85, 91, 101, 102, 105, 109, 110, 114, 129, 133, 138, 141, 151, 154, 158, 165, 170, 174, 181, 182, 193, 194, 199, 201, 213, 226, 229, 230, 235, 241, 246, 253, 258, 265, 271, 273
Offset: 1

Views

Author

Leroy Quet, Oct 06 2005

Keywords

Examples

			10 is the smallest squarefree number greater than the 4th prime, 7. So a(4) = 10.
From _Gus Wiseman_, Dec 07 2024: (Start)
The first number line below shows the squarefree numbers. The second shows the primes:
--1--2--3-----5--6--7-------10-11----13-14-15----17----19----21-22-23-------26--
=====2==3=====5=====7==========11====13==========17====19==========23===========
(End)

Links

Crossrefs

Cf. A061400, A112928, A112929, A112930.
Restriction of A067535, differences A378087.
The unrestricted opposite is A070321, differences A378085.
The opposite is A112925, differences A378038.
Subtracting prime(n) from each term gives A240474, opposite A240473.
For nonsquarefree we have A377783, restriction of A120327.
The nonsquarefree differences are A377784, restriction of A378039.
First differences are A378037.
For perfect power we have A378249, A378617, A378250, A378251.
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. A007674, A013928, A053797, A053806, A072284, A073247, A175216, A280892, A345531, A378032, A378618.

Programs

  • Maple
    with(numtheory): a:=proc(n) local p,B,j: p:=ithprime(n): B:={}: for j from p+1 to p+20 do if abs(mobius(j))>0 then B:=B union {j} else B:=B fi od: B[1] end: seq(a(m),m=1..75); # Emeric Deutsch, Oct 10 2005
  • Mathematica
    Do[k = Prime[n] + 1; While[ !SquareFreeQ[k], k++ ]; Print[k], {n, 1, 100}] (* Ryan Propper, Oct 10 2005 *)
With[{k = 120}, Table[SelectFirst[Range[Prime@ n + 1, Prime@ n + k], SquareFreeQ], {n, 58}]] (* Michael De Vlieger, Aug 16 2017 *)
  • PARI
    a(n,p=prime(n))=while(!issquarefree(p++),); p \\ Charles R Greathouse IV, Aug 16 2017

Formula

a(n) = prime(n) + A240474(n). - Gus Wiseman, Dec 07 2024

Extensions

More terms from Ryan Propper and Emeric Deutsch, Oct 10 2005

A299763 a(n) = 1 + A182986(n).

Original entry on oeis.org

1, 3, 4, 6, 8, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 54, 60, 62, 68, 72, 74, 80, 84, 90, 98, 102, 104, 108, 110, 114, 128, 132, 138, 140, 150, 152, 158, 164, 168, 174, 180, 182, 192, 194, 198, 200, 212, 224, 228, 230, 234, 240, 242, 252, 258, 264, 270, 272, 278, 282, 284
Offset: 1

Views

Author

Omar E. Pol, Mar 14 2018

Keywords

Comments

Are these the indices of the rows of A299762 where there is a record?

Links

Crossrefs

First differences are in A054541.
Cf. A008864, A028815, A182986, A299762.
Essentially the same as A008864, A028815, A055670, A135731, A175216.

Programs

Formula

a(n) = A028815(n-1) - [n=1].
a(n) = A008864(n-1) for n >= 2, with a(1) = 1.

A175218 The third nonprimes after the primes.

Original entry on oeis.org

8, 8, 9, 10, 15, 16, 21, 22, 26, 33, 34, 40, 45, 46, 50, 56, 63, 64, 70, 75, 76, 82, 86, 92, 100, 105, 106, 111, 112, 116, 130, 134, 141, 142, 153, 154, 160, 166, 170, 176, 183, 184, 195, 196, 201, 202, 214, 226, 231, 232, 236, 243, 244, 254, 260, 266, 273, 274
Offset: 1

Views

Author

Jaroslav Krizek, Mar 06 2010

Keywords

Examples

			prime(49) = 227, prime(50) = 229, therefore (228=1st, 230=2nd nonprime), 231 = a(49). - _Georg Fischer_, Oct 06 2024

Crossrefs

Cf. A175216, A175217, A175219, A175220.

Programs

  • PARI
    a(n) = if(n<=2, 8, prime(n) + 2 + isprime(prime(n)+2) + 1) /* Georg Fischer, Oct 06 2024 */

Formula

For n>2: a(n) = A000040(n) + 2 + A010051(A000040(n) + 2) + 1. - Reinhard Zumkeller, Mar 29 2010 [corrected by Georg Fischer, Oct 06 2024]

Extensions

More terms from Reinhard Zumkeller, Mar 29 2010
a(49), a(52) and a(57) corrected by Georg Fischer, Oct 06 2024
Showing 1-4 of 4 results.

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