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 1-5 of 5 results.

A081213 Let r(n,k) = if k=0 then n, else r(A081210(n),k-1), then a(n)=r(n, A081212(n)).

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 7, 7, 10, 11, 7, 13, 14, 15, 15, 17, 14, 19, 15, 21, 22, 23, 21, 23, 26, 26, 21, 29, 30, 31, 31, 33, 34, 35, 21, 37, 38, 39, 35, 41, 42, 43, 33, 35, 46, 47, 35, 47, 46, 51, 39, 53, 39, 55, 47, 57, 58, 59, 35, 61, 62, 47, 62, 65, 66, 67, 51, 69, 70, 71, 47, 73
Offset: 1

Views

Author

Reinhard Zumkeller, Mar 10 2003

Keywords

Comments

A081210(a(n)) = a(n).
Different from A081211.
a(n) = A081211(n) for n<84 = A131072(1); a(A131072(n)) <> A081211(A131072(n)). - Reinhard Zumkeller, Jun 13 2007

Links

Crossrefs

Cf. A081210, A081211, A081212, A081214.
Cf. A131072.

Programs

  • Maple
    A081212r := proc(n,k)
    option remember ;
    if k =0 then
        n;
    else
        procname(A081210(n),k-1) ;
    end if;
end proc:
A081212 := proc(n)
    local i ;
    for i from 0 do
        if A081212r(n,i) = A081212r(n,i+1) then
            return i ;
        end if;
    end do:
end proc:
A081213 := proc(n)
    A081212r(n,A081212(n)) ;
end proc:
seq(A081213(n),n=1..84) ; # R. J. Mathar, May 25 2023
  • Mathematica
    gsf[n_] := For[k = n, True, k--, If[SquareFreeQ[k], Return[k]]];
A081210[n_] := (cnt++; Times @@ gsf /@ Power @@@ FactorInteger[n]);
A081212[n_] := (cnt = 0; FixedPoint[A081210, n]; cnt - 1);
r[n_, k_] := r[n, k] = If[k == 0, n, r[A081210[n], k - 1]];
a[n_] := r[n, A081212[n]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Sep 12 2023 *)

A081214 Least m such that A081212(m) = n.

Original entry on oeis.org

1, 4, 12, 84, 3948, 88320, 815424, 14652150
Offset: 0

Views

Author

Reinhard Zumkeller, Mar 10 2003

Keywords

Crossrefs

Cf. A081210, A081212.

Programs

  • Maple
    gsqfree := proc(n) local resul ; resul := n ; while not numtheory[issqrfree](resul) do resul := resul-1 ; od ; RETURN(resul) ; end: A081210 := proc(n) option remember ; local pfd,resul,p ; if n = 1 then RETURN(1) ; else pfd := ifactors(n)[2] ; resul := 1 ; for p from 1 to nops(pfd) do resul := resul*gsqfree( op(1,op(p,pfd))^op(2,op(p,pfd))) ; od ; fi ; end: r := proc(n,k) option remember ; if k= 0 then n ; else r(A081210(n),k-1) ; fi ; end: A081212 := proc(n) local i; i := 0 ; while r(n,i) <> r(n,i+1) do i := i+1 ; od ; RETURN(i) ; end: A081214 := proc() local a,m,h ; a :=[seq(-1,i=1..40)] ; for m from 1 to 8000000 do h := A081212(m) ; if h+1 <= nops(a) then if op(h+1,a) = -1 then a := subsop(h+1=m,a) ; print(a) ; fi ; fi ; od ; RETURN(a) ; end: A081214() ; # R. J. Mathar, Apr 04 2007
  • Mathematica
    gsf[n_] := gsf[n] = For[k = n, True, k--, If[SquareFreeQ[k], Return[k]]];
A081210[n_] := A081210[n] = Times @@ gsf /@ Power @@@ FactorInteger[n];
A081212[n_] := A081212[n] = Module[{cnt = 0}, FixedPoint[(cnt++; A081210[#])&, n]; cnt - 1];
a[n_] := a[n] = For[m = 1, True, m++, If[A081212[m] == n, Return[m]]];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 7}] (* Jean-François Alcover, Nov 27 2023 *)

Extensions

More terms from R. J. Mathar, Apr 04 2007
One more term from Jean-François Alcover, Nov 27 2023

A081210 In prime factorization of n replace each prime power p^e with the greatest squarefree number <= p^e.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 7, 7, 10, 11, 9, 13, 14, 15, 15, 17, 14, 19, 15, 21, 22, 23, 21, 23, 26, 26, 21, 29, 30, 31, 31, 33, 34, 35, 21, 37, 38, 39, 35, 41, 42, 43, 33, 35, 46, 47, 45, 47, 46, 51, 39, 53, 52, 55, 49, 57, 58, 59, 45, 61, 62, 49, 62, 65, 66, 67, 51, 69, 70, 71, 49, 73
Offset: 1

Views

Author

Reinhard Zumkeller, Mar 10 2003

Keywords

Links

Crossrefs

Cf. A005117, A070321, A081212, A081213, A081214.

Programs

  • Maple
    A081210 := proc(n)
    local a,pe;
    a :=1 ;
    for pe in ifactors(n)[2] do
        a := a*A070321(op(1,pe)^op(2,pe)) ;
    end do:
    a ;
end proc:
seq(A081210(n),n=1..100) ; # R. J. Mathar, May 25 2023
  • Mathematica
    gsf[n_] := For[k = n, True, k--, If[ SquareFreeQ[k], Return[k]]]; a[n_] := Times @@ gsf /@ Power @@@ FactorInteger[n]; Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Mar 27 2013 *)
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] == 1, f[i,1], my(k = f[i,1]^f[i,2]); while(!issquarefree(k), k--); k));} \\ Amiram Eldar, Jun 09 2025

Formula

Multiplicative with a(p^e) = A070321(p^e), p prime.
a(n) <= n and a(n) = n iff n is squarefree (A005117).
A081211(n) = a(a(n)), see A081212, A081213 and A081214 for iterations until a fixed point is reached.

A081211 a(n) = A081210(A081210(n)).

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 7, 7, 10, 11, 7, 13, 14, 15, 15, 17, 14, 19, 15, 21, 22, 23, 21, 23, 26, 26, 21, 29, 30, 31, 31, 33, 34, 35, 21, 37, 38, 39, 35, 41, 42, 43, 33, 35, 46, 47, 35, 47, 46, 51, 39, 53, 39, 55, 47, 57, 58, 59, 35, 61, 62, 47, 62, 65, 66, 67, 51, 69, 70, 71, 47, 73
Offset: 1

Views

Author

Reinhard Zumkeller, Mar 10 2003

Keywords

Comments

a(n) = r(n,2), where r is defined as in A081212, A081213.
Different from A081213 (see example).

Examples

			Recall that A081210 = (in prime factorization of n: replace each prime power p^e = by the greatest squarefree number <= p^e).
Consider n = 84 = 2*2*3*7,
A081210(84) = 3*3*7 = 63,
A081210(A081210(84)) = A081210(63) = 7*7 = 49 = a(84),
A081210(A081210(A081210(84))) = A081210(A081210(63)) = A081210(49) = 47,
A081212(49) = 3 as A081210(47) = 47 hence A081213(84) = 47,
Therefore a(84) <> A081213(84), 49 <> 47.

Links

Crossrefs

Cf. A081210, A081213, A131072.

Programs

Formula

a(n) = A081213(n) for n<84 = A131072(1); a(A131072(n)) <> A081213(A131072(n)). - Reinhard Zumkeller, Jun 13 2007

A131072 Numbers m such that A081211(m) <> A081213(m).

Original entry on oeis.org

84, 336, 420, 924, 1092, 1428, 1452, 1596, 1680, 1840, 1932, 2057, 2100, 2436, 2604, 2625, 2632, 2961, 3000, 3108, 3384, 3444, 3468, 3500, 3528, 3612, 3696, 3948, 4114, 4368, 4452, 4500, 4620, 4956, 5124, 5250, 5376, 5460, 5520, 5544, 5628, 5712, 5808
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 13 2007

Keywords

Comments

Subsequence of A013929;
A081212(a(n)) > 2; A081211(a(n)) <> A081213(a(n));
suggested by Andrew S. Plewe regarding the equality of initial terms of A081211 and A081213.
Showing 1-5 of 5 results.

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