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.

A065330 a(n) = max { k | gcd(n, k) = k and gcd(k, 6) = 1 }.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 7, 1, 1, 5, 11, 1, 13, 7, 5, 1, 17, 1, 19, 5, 7, 11, 23, 1, 25, 13, 1, 7, 29, 5, 31, 1, 11, 17, 35, 1, 37, 19, 13, 5, 41, 7, 43, 11, 5, 23, 47, 1, 49, 25, 17, 13, 53, 1, 55, 7, 19, 29, 59, 5, 61, 31, 7, 1, 65, 11, 67, 17, 23, 35, 71, 1, 73, 37, 25, 19, 77, 13, 79, 5, 1
Offset: 1

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Author

Reinhard Zumkeller, Oct 29 2001

Keywords

Comments

Bennett, Filaseta, & Trifonov show that if n > 8 then a(n^2 + n) > n^0.285. - Charles R Greathouse IV, May 21 2014

Examples

			a(30) = 5.

Links

Crossrefs

Cf. A065331, A000265, A038502, A165725.

Programs

  • Haskell
    a065330 = a038502 . a000265  -- Reinhard Zumkeller, Jul 06 2011
  • Magma
    [n div Gcd(n, 6^n): n in [1..100]]; // Vincenzo Librandi, Feb 09 2016
  • Maple
    A065330 := proc(n)
    local a,f,p,e ;
    a := 1 ;
    for f in ifactors(n)[2] do
        p := op(1,f) ;
        e := op(2,f) ;
        if p > 3 then
            a := a*p^e ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Jul 12 2012
with(padic): a := n -> n/(2^ordp(n, 2)*3^ordp(n, 3));
seq(a(n), n=1..81); # Peter Luschny, Mar 25 2014
  • Mathematica
    f[n_] := Times @@ (First@#^Last@# & /@ Select[FactorInteger@n, First@# != 2 && First@# != 3 &]); Array[f, 81] (* Robert G. Wilson v, Aug 18 2006 *)
f[n_]:=Denominator[6^n/n];Array[f,100] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2011 *)
Table[n / GCD[n, 6^n], {n, 100}] (* Vincenzo Librandi, Feb 09 2016 *)
  • PARI
    a(n)=if(n<2,1,if(n%2,if(n%3,n,a(n/3)),a(n/2))) \\ Benoit Cloitre, Jun 04 2007
  • PARI
    a(n)=n\gcd(n,6^n) \\ Not very efficient, but simple. Stanislav Sykora, Feb 08 2016
  • PARI
    a(n)=n>>valuation(n,2)/3^valuation(n,3) \\ Charles R Greathouse IV, Mar 31 2016

Formula

a(n) * A065331(n) = n.
Multiplicative with a(2^e)=1, a(3^e)=1, a(p^e)=p^e, p>3. - Vladeta Jovovic, Nov 02 2001
A106799(n) = A001222(a(n)). - Reinhard Zumkeller, May 19 2005
a(1)=1; then a(2n)=a(n), a(2n+1)=a((2n+1)/3) if 2n+1 is divisible by 3, a(2n+1)=2n+1 otherwise. - Benoit Cloitre, Jun 04 2007
Dirichlet g.f. zeta(s-1)*(1-2^(1-s))*(1-3^(1-s))/ ( (1-2^(-s))*(1-3^(-s)) ). - R. J. Mathar, Jul 04 2011
a(n) = A038502(A000265(n)). - Reinhard Zumkeller, Jul 06 2011
a(n) = n/GCD(n,6^n). - Stanislav Sykora, Feb 08 2016
Sum_{k=1..n} a(k) ~ (1/4) * n^2. - Amiram Eldar, Oct 22 2022

A169611 Number of prime divisors of n that are not greater than 3, counted with multiplicity.

Original entry on oeis.org

0, 1, 1, 2, 0, 2, 0, 3, 2, 1, 0, 3, 0, 1, 1, 4, 0, 3, 0, 2, 1, 1, 0, 4, 0, 1, 3, 2, 0, 2, 0, 5, 1, 1, 0, 4, 0, 1, 1, 3, 0, 2, 0, 2, 2, 1, 0, 5, 0, 1, 1, 2, 0, 4, 0, 3, 1, 1, 0, 3, 0, 1, 2, 6, 0, 2, 0, 2, 1, 1, 0, 5, 0, 1, 1, 2, 0, 2, 0, 4, 4, 1, 0, 3, 0, 1, 1, 3, 0, 3, 0, 2, 1, 1, 0, 6, 0, 1, 2, 2, 0, 2, 0, 3, 1
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Dec 03 2009

Keywords

Links

Crossrefs

Cf. A001222, A106799.
Cf. A007814, A007949, A065331.

Programs

  • Haskell
    a169611 = a001222 . a065331  -- Reinhard Zumkeller, Nov 19 2015
  • Maple
    A169611 := proc(n) local f; a := 0 ; for f in ifactors(n)[2] do if op(1,f) <= 3 then a := a+op(2,f) ; end if; end do: return a; end proc: seq(A169611(n),n=1..100) ; # R. J. Mathar, Dec 04 2009
  • Mathematica
    f[n_] := Plus @@ Last /@ Select[ FactorInteger@ n, 1 < #[[1]] < 4 &]; Array[f, 105] (* Robert G. Wilson v, Dec 19 2009 *)
  • PARI
    A169611(n)=valuation(n,2)+valuation(n,3)  \\ M. F. Hasler, Aug 24 2012

Formula

a(n) = A001222(n) - A106799(n).
a(n) = A007814(n) + A007949(n). - R. J. Mathar, Dec 04 2009
a(n) = A001222(A065331(n)). - Reinhard Zumkeller, Nov 19 2015
Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3/2. - Amiram Eldar, Jan 16 2022

Extensions

Definition corrected by M. F. Hasler, Aug 24 2012

A375670 The maximum exponent in the prime factorization of the largest 5-rough divisor of n.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 2, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 2, 2, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 2, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 1, 2, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Amiram Eldar, Aug 23 2024

Keywords

Comments

First differs from A106799 at n = 35.
The largest exponent among the exponents of the primes that are larger than 3 in the prime factorization of n.

Links

Crossrefs

Cf. A003586, A007310 (5-rough numbers), A051903, A065330, A106799, A375669.

Programs

  • Mathematica
    a[n_] := Module[{m = n / Times@@({2,3}^IntegerExponent[n,{2,3}])}, If[m == 1, 0, Max[FactorInteger[m][[;; , 2]]]]]; Array[a, 100]
  • PARI
    a(n) = {my(m = n >> valuation(n, 2)/3^valuation(n, 3)); if(m == 1, 0,vecmax(factor(m)[,2]));}

Formula

a(n) = A051903(A065330(n)).
a(n) = 0 if and only if n is a 3-smooth number (A003586).
a(n) = 1 if and only if n is a product of a squarefree 5-rough number larger than 1 and a 3-smooth number.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} k * d(k) = 1.1034178389191320571029... , where d(k) is the asymptotic density of the occurrences of k in this sequence: d(1) = 3/(2*zeta(2)), and d(k) = (1/zeta(k+1)) / ((1-1/2^(k+1))*(1-1/3^(k+1))) - (1/zeta(k)) / ((1-1/2^k)*(1-1/3^k)) for k >= 2.
In general, the asymptotic mean of the maximum exponent in the prime factorization of the largest p-rough divisor of n is Sum_{k>=1} k * d(k), where d(1) = 1/(zeta(2) * f(p, 2)), d(k) = 1/(zeta(k+1) * f(p, k+1)) - 1/(zeta(k) * f(p, k)) for k >= 2, and f(p, m) = Product_{q prime < p} (1-1/q^m).
Showing 1-3 of 3 results.

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