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 1-3 of 3 results.

A076259 Gaps between squarefree numbers: a(n) = A005117(n+1) - A005117(n).

Original entry on oeis.org

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

Views

Author

Reinhard Zumkeller, Oct 03 2002

Keywords

Comments

This sequence is unbounded, as a simple consequence of the Chinese remainder theorem. - Thomas Ordowski, Jul 22 2015
Conjecture: lim sup_{n->oo} a(n)/log(A005117(n)) = 1/2. - Thomas Ordowski, Jul 23 2015 [Note: this conjecture is equivalent to lim sup a(n)/log n = 1/2. - Charles R Greathouse IV, Dec 05 2024]
a(n) = 1 infinitely often since the density of the squarefree numbers, 6/Pi^2, is greater than 1/2. In particular, at least 2 - Pi^2/6 = 35.5...% of the terms are 1. - Charles R Greathouse IV, Jul 23 2015
From Amiram Eldar, Mar 09 2021: (Start)
The asymptotic density of the occurrences of 1 in this sequence is density(A007674)/density(A005117) = A065474/A059956 = 0.530711... (A065469).
The asymptotic density of the occurrences of 2 in this sequence is (density(A069977)-density(A007675))/density(A005117) = (A065474-A206256)/A059956 = 0.324294... (End)

Examples

			As 24 = 3*2^3 and 25 = 5^2, the next squarefree number greater A005117(16) = 23 is A005117(17) = 26, therefore a(16) = 26-23 = 3.

Links

Crossrefs

Cf. A005117, A013661, A020753 (records), A020754, A076260.
Cf. A007674, A007675, A059956, A065469, A065474, A069977, A206256.

Programs

  • Haskell
    a076259 n = a076259_list !! (n-1)
a076259_list = zipWith (-) (tail a005117_list) a005117_list
-- Reinhard Zumkeller, Aug 03 2012
  • Maple
    A076259 := proc(n) A005117(n+1)-A005117(n) ; end proc: # R. J. Mathar, Jan 09 2013
  • Mathematica
    Select[Range[200], SquareFreeQ] // Differences (* Jean-François Alcover, Mar 10 2019 *)
  • PARI
    t=1; for(n=2,1e3, if(issquarefree(n), print1(n-t", "); t=n)) \\ Charles R Greathouse IV, Jul 23 2015
  • Python
    from math import isqrt
from sympy import mobius
def A076259(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    r, k = n+1, f(n+1)+1
    while r != k:
        r, k = k, f(k)+1
    return int(r-m) # Chai Wah Wu, Aug 15 2024

Formula

Asymptotic mean: lim_{n->oo} (1/n) Sum_{k=1..n} a(k) = Pi^2/6 (A013661). - Amiram Eldar, Oct 21 2020
a(n) < n^(1/5) for large enough n by a result of Pandey. (The constant Pi^2/6 can be absorbed by any eta > 0.) - Charles R Greathouse IV, Dec 04 2024

A070321 Greatest squarefree number <= n.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 7, 7, 10, 11, 11, 13, 14, 15, 15, 17, 17, 19, 19, 21, 22, 23, 23, 23, 26, 26, 26, 29, 30, 31, 31, 33, 34, 35, 35, 37, 38, 39, 39, 41, 42, 43, 43, 43, 46, 47, 47, 47, 47, 51, 51, 53, 53, 55, 55, 57, 58, 59, 59, 61, 62, 62, 62, 65, 66, 67, 67, 69, 70, 71, 71
Offset: 1

Views

Author

Benoit Cloitre, May 11 2002

Keywords

Comments

a(n) = Max( core(k) : k=1,2,3,...,n ) where core(x) is the squarefree part of x (the smallest integer such that x*core(x) is a square).

Examples

			From _Gus Wiseman_, Dec 10 2024: (Start)
The squarefree numbers <= n are the following columns, with maxima a(n):
  1  2  3  3  5  6  7  7  7  10  11  11  13  14  15  15
     1  2  2  3  5  6  6  6  7   10  10  11  13  14  14
        1  1  2  3  5  5  5  6   7   7   10  11  13  13
              1  2  3  3  3  5   6   6   7   10  11  11
                 1  2  2  2  3   5   5   6   7   10  10
                    1  1  1  2   3   3   5   6   7   7
                             1   2   2   3   5   6   6
                                 1   1   2   3   5   5
                                         1   2   3   3
                                             1   2   2
                                                 1   1
(End)

Links

Crossrefs

Cf. A007947, A076260.
Cf. A081217, A081218, A081210.
The distinct terms are A005117 (the squarefree numbers).
The opposite version is A067535, differences A378087.
The run-lengths are A076259.
Restriction to the primes is A112925; see A378038, A112926, A378037.
For nonsquarefree we have A378033; see A120327, A378036, A378032, A377783.
First differences are A378085.
Subtracting each term from n gives A378619.
A013929 lists the nonsquarefree numbers, differences A078147.
A061398 counts squarefree numbers between primes, zeros A068360.
A061399 counts nonsquarefree numbers between primes, zeros A068361.
Cf. A007674, A013928, A053797, A053806, A072284, A073247, A112929, A240473.

Programs

  • Maple
    A070321 := proc(n)
    local a;
    for a from n by -1 do
        if issqrfree(a) then
            return a;
        end if;
    end do:
end proc:
seq(A070321(n),n=1..100) ; # R. J. Mathar, May 25 2023
  • Mathematica
    a[n_] :=For[ k = n, True, k--, If[ SquareFreeQ[k], Return[k]]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Mar 27 2013 *)
gsfn[n_]:=Module[{k=n},While[!SquareFreeQ[k],k--];k]; Array[gsfn,80] (* Harvey P. Dale, Mar 27 2013 *)
  • PARI
    a(n) = while (! issquarefree(n), n--); n; \\ Michel Marcus, Mar 18 2017
  • Python
    from itertools import count
from sympy import factorint
def A070321(n): return next(m for m in count(n,-1) if max(factorint(m).values(),default=0)<=1) # Chai Wah Wu, Dec 04 2024

Formula

a(n) = n - o(n^(1/5)) by a result of Pandey. - Charles R Greathouse IV, Dec 04 2024
a(n) = A005117(A013928(n+1)). - Ridouane Oudra, Jul 26 2025

Extensions

New description from Reinhard Zumkeller, Oct 03 2002

A067535 Smallest squarefree number >= n.

Original entry on oeis.org

1, 2, 3, 5, 5, 6, 7, 10, 10, 10, 11, 13, 13, 14, 15, 17, 17, 19, 19, 21, 21, 22, 23, 26, 26, 26, 29, 29, 29, 30, 31, 33, 33, 34, 35, 37, 37, 38, 39, 41, 41, 42, 43, 46, 46, 46, 47, 51, 51, 51, 51, 53, 53, 55, 55, 57, 57, 58, 59, 61, 61, 62
Offset: 1

Views

Author

Reinhard Zumkeller, Jan 27 2002

Keywords

Links

Crossrefs

Cf. A005117, A070321, A076260, A081221.

Programs

  • Maple
    A067535 := proc(n)
    for a from n do
        if issqrfree(a) then
            return a ;
        end if;
    end do:
end proc:
seq(A067535(n),n=1..100) ; # R. J. Mathar, May 31 2024
  • Mathematica
    Table[k = n; While[! SquareFreeQ@ k, k++]; k, {n, 62}] (* Michael De Vlieger, Mar 18 2017 *)
  • PARI
    a(n) = while (! issquarefree(n), n++); n; \\ Michel Marcus, Mar 18 2017
  • Python
    from itertools import count
from sympy import factorint
def A067535(n): return next(m for m in count(n) if max(factorint(m).values(),default=0)<=1) # Chai Wah Wu, Dec 04 2024

Formula

a(n) = n + A081221(n). - Amiram Eldar, Oct 10 2023
Showing 1-3 of 3 results.

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