A131072 -id:A131072 - cp's OEIS frontend

A081212 Let r(n,k) = if k=0 then n else r(A081210(n),k-1), then a(n) = Min{i:r(n,i) = r(n,i+1)}.

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 1, 1, 0, 1, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 2, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 3, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0

Offset: 1

Reinhard Zumkeller, Mar 10 2003

nonn

Number of times A081210 is to be applied to n until a fixed-point is reached, A081213(n)=r(n,a(n)).

A081211(n) = A081213(n) iff a(n) <= 2; a(A131072(n)) > 2. - Reinhard Zumkeller, Jun 13 2007

R. Zumkeller, Table of n, a(n) for n = 1..10000

gsf[n_] := For[k = n, True, k--, If[SquareFreeQ[k], Return[k]]];
A081210[n_] := Times @@ gsf /@ Power @@@ FactorInteger[n];
a[n_] := Module[{cnt = 0}, FixedPoint[(cnt++; A081210[#])&, n]; cnt-1];
Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Mar 27 2013, updated Nov 27 2023 *)

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

Reinhard Zumkeller, Mar 10 2003

nonn

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

Different from A081213 (see example).

			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.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A081210, A081213, A131072.

A081211 := proc(n)
    A081210(A081210(n)) ;
end proc:
seq(A081211(n),n=1..84) ; # R. J. Mathar, May 25 2023

gsf[n_] := For[k = n, True, k--, If[SquareFreeQ[k], Return[k]]];
A081210[n_] := Times @@ gsf /@ Power @@@ FactorInteger[n];
a[n_] := A081210[A081210[n]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Sep 12 2023 *)

A081210(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));}
a(n) = A081210(A081210(n)); \\ Amiram Eldar, Jun 09 2025

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

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

Offset: 1

Reinhard Zumkeller, Mar 10 2003

nonn

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

R. Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A081210, A081211, A081212, A081214.

Cf. A131072.

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

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 *)

cp's OEIS Frontend

A081212 Let r(n,k) = if k=0 then n else r(A081210(n),k-1), then a(n) = Min{i:r(n,i) = r(n,i+1)}.

Views

Author

Keywords

Comments

Links

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica