A097249 - OEIS

%I #15 Nov 18 2021 11:06:13

%S 0,0,0,1,0,0,0,1,1,0,0,2,0,0,0,2,0,1,0,1,0,0,0,2,1,0,1,1,0,0,0,2,0,0,

%T 0,1,0,0,0,1,0,0,0,1,2,0,0,2,1,1,0,1,0,1,0,1,0,0,0,3,0,0,1,2,0,0,0,1,

%U 0,0,0,1,0,0,1,1,0,0,0,3,2,0,0,2,0,0,0,1,0,2,0,1,0,0,0,2,0,1,1,1,0,0,0,1,0

%N a(n) is the number of times we must iterate A097246, starting at n, before the result is squarefree.

%C a(n) = Min{k: r(n,k)=r(n,k+1)}, where r(n,k)=A097246(r(n,k-1)), r(n,0)=n;

%C a(A005117(n))=0; a(A097250(n))=n and a(m)<n for m < A097250(n).

%H Antti Karttunen, <a href="/A097249/b097249.txt">Table of n, a(n) for n = 1..65537</a>

%F If A008966(n) = 1 [when n is in A005117], a(n) = 0, otherwise a(n) = 1 + a(A097246(n)). - _Antti Karttunen_, Jul 29 2018

%t f[n_] := Product[{p, e} = pe; NextPrime[p]^Quotient[e, 2] p^Mod[e, 2], {pe, FactorInteger[n]}];

%t a[n_] := (NestWhileList[f, n, !SquareFreeQ[#]&] // Length) - 1;

%t Array[a, 105] (* _Jean-François Alcover_, Nov 18 2021 *)

%o (PARI)

%o A097246(n) = { my(f=factor(n)); prod(i=1, #f~, (nextprime(f[i,1]+1)^(f[i,2]\2))*((f[i,1])^(f[i,2]%2))); };

%o A097249(n) = if(issquarefree(n),0,1+A097249(A097246(n))); \\ _Antti Karttunen_, Jul 29 2018

%Y Cf. A005117, A097246, A097248, A097250, A277899.

%K nonn

%O 1,12

%A _Reinhard Zumkeller_, Aug 03 2004

%E Edited by _Sam Alexander_, Jan 05 2005