A081211 -id:A081211 - cp's OEIS frontend

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

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

Reinhard Zumkeller, Jun 13 2007

nonn

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.

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

Reinhard Zumkeller, Mar 10 2003

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

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

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

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 *)

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

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.

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)}.

Original entry on oeis.org

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 *)

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

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 *)

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

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

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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)}.

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

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