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.

A204315 Numbers j such that floor(j^(1/4)) divides j.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126
Offset: 1

Views

Author

Benoit Cloitre, Jan 14 2012

Keywords

Examples

			26 is a term as floor(26^(1/4)) = 2 divides 26. - _David A. Corneth_, Oct 04 2023

Links

Crossrefs

Cf. A079645, A228318.

Programs

  • Maple
    isA204315 := proc(n)
    if modp(n,floor(root[4](n))) = 0 then
        true ;
    else
        false ;
    fi ;
end proc:
for n from 1 to 130 do
    if isA204315(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Sep 10 2017
  • Mathematica
    Select[Range[150],Mod[#,Floor[Surd[#,4]]]==0&] (* Harvey P. Dale, Oct 04 2023 *)
  • PARI
    a(n) = {my(k = 0, t = 0); while(t < n, k++; t = 4*k^3/3 + 5*k^2 + 26*k/3); (k+1)^4 - 1 - k * (t - n)} \\ David A. Corneth, Oct 06 2023
  • PARI
    first(n) = {my(res = vector(n), t = 0); for(i = 1, oo, forstep(j = i^4, (i + 1)^4 - 1, i, t++; if(t > n, return(res)); res[t] = j))} \\ David A. Corneth, Oct 06 2023

Formula

Let f(x) = 4*x^3/3 + 5*x^2 + 26*x/3 and let k be the smallest integer x such that f(x) >= n. Then a(n) = (k+1)^4 - 1 - k * (f(k) - n). - David A. Corneth, Oct 06 2023

A228319 The hyper-Wiener index of the graph obtained by applying Mycielski's construction to the star graph K(1,n).

Original entry on oeis.org

20, 45, 82, 131, 192, 265, 350, 447, 556, 677, 810, 955, 1112, 1281, 1462, 1655, 1860, 2077, 2306, 2547, 2800, 3065, 3342, 3631, 3932, 4245, 4570, 4907, 5256, 5617, 5990, 6375, 6772, 7181, 7602, 8035, 8480, 8937, 9406, 9887, 10380
Offset: 1

Views

Author

Emeric Deutsch, Aug 27 2013

Keywords

References

  • D. B. West, Introduction to Graph Theory, 2nd ed., Prentice-Hall, NJ, 2001, p. 205.

Links

Crossrefs

Cf. A228318.

Programs

  • Maple
    a := proc (n) options operator, arrow: 6*n^2+7*n+7 end proc: seq(a(n), n = 1 .. 42);
  • PARI
    a(n)=6*n^2+7*n+7 \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = 6*n^2 + 7*n + 7.
G.f.: x*(20-15*x+7*x^2)/(1-x)^3.
The Hosoya-Wiener polynomial is (4*n+1)*t + (2*n^2 + n + 2)*t^2.
From Elmo R. Oliveira, Nov 15 2024: (Start)
E.g.f.: exp(x)*(6*x^2 + 13*x + 7) - 7.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

A228599 The Wiener index of the graph obtained by applying Mycielski's construction to the rooted tree having Matula-Goebel number n.

Original entry on oeis.org

5, 15, 33, 33, 62, 62, 59, 59, 103, 103, 103, 99, 99, 99, 156, 93, 99, 151, 93, 152, 152, 156, 151, 144, 221, 151, 215, 147, 152, 216, 156, 135, 221, 152, 217, 207, 144, 144, 216, 209, 151, 211, 147, 217, 292, 215, 216, 197, 213, 293, 217, 211
Offset: 1

Views

Author

Emeric Deutsch, Aug 29 2013

Keywords

Comments

The Matula-Goebel number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T.
a(2^n) = A228318(n). Indeed, the rooted tree corresponding to the Matula-Goebel number 2^n is the star graph K(1,n).
a(A007097(n)) = A228321(n). Indeed, A007097(n) for n=1,2,... yields the primeth recurrence sequence (A007097(1)=2, A007097(n+1)=A007097(n)-th prime; first few terms are 2,3,5,11,31,127,709). The corresponding rooted trees are the path trees on n+1 vertices.

References

  • D. B. West, Introduction to Graph Theory, 2nd ed., Prentice-Hall, NJ, 2001, p. 205.

Links

Crossrefs

Cf. A061775, A196050, A196059, A228318, A007097, A228321

Programs

  • Maple
    with(numtheory): V := proc (n) local u, v: u := proc (n) options operator, arrow: op(1, factorset(n)) end proc: v := proc (n) options operator, arrow: n/u(n) end proc: if n = 1 then 1 elif isprime(n) then 1+V(pi(n)) else V(u(n))+V(v(n))-1 end if end proc: WP := proc (n) local r, s, R: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: R := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then sort(expand(x*R(pi(n))+x)) else sort(expand(R(r(n))+R(s(n)))) end if end proc: if n = 1 then 0 elif bigomega(n) = 1 then sort(expand(WP(pi(n))+x*R(pi(n))+x)) else sort(expand(WP(r(n))+WP(s(n))+R(r(n))*R(s(n)))) end if end proc: p2 := proc (n) options operator, arrow: coeff(WP(n), x, 2) end proc: p3 := proc (n) options operator, arrow: coeff(WP(n), x, 3) end proc: a := proc (n) options operator, arrow: 6*V(n)^2-8*V(n)+7-4*p2(n)-p3(n) end proc: seq(a(n), n = 1 .. 80);

Formula

In Balakrishnan et al. one proves that the Wiener index of the Mycielskian of a connected graph G is 6V^2 - V - 7E - 4p(2) - p(3), where V is number of vertices of G, E is number of edges in G, and p(i) is number of pairs of vertices in G which are at distance i. For the rooted tree with Matula-Goebel number n these quantities can be found in A061775, A196050, and A196059.
Showing 1-3 of 3 results.

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