cp's OEIS Frontend

A327503 Number of steps to reach a fixed point starting with n and repeatedly taking the quotient by the maximum divisor that is 1 or not a perfect power (A327501, A327502).

%I A327503 #20 Apr 03 2022 22:53:39
%S A327503 0,1,1,2,1,1,1,3,2,1,1,1,1,1,1,4,1,1,1,1,1,1,1,1,2,1,3,1,1,1,1,5,1,1,
%T A327503 1,2,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,6,1,1,1,1,
%U A327503 1,1,1,1,1,1,1,1,1,1,1,1,4,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1
%N A327503 Number of steps to reach a fixed point starting with n and repeatedly taking the quotient by the maximum divisor that is 1 or not a perfect power (A327501, A327502).
%C A327503 First differs from A052409 and A158378 at a(216) = 2, A052409(216) = A158378(216) = 3.
%C A327503 A multiset is aperiodic if its multiplicities are relatively prime. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). Heinz numbers of aperiodic multisets are numbers that are not perfect powers (A007916).
%C A327503 a(n) does not depend only on the prime signature of n. A351948 gives the positions where a(A046523(n)) <> a(n). n = 125000 is the first time this happens, see the examples. - _Antti Karttunen_, Apr 03 2022
%H A327503 Antti Karttunen, <a href="/A327503/b327503.txt">Table of n, a(n) for n = 1..65537</a>
%H A327503 Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vSX9dPMGJhxB8rOknCGvOs6PiyhupdWNpqLsnphdgU6MEVqFBnWugAXidDhwHeKqZe_YnUqYeGOXsOk/pub">Sequences counting and encoding certain classes of multisets</a>
%F A327503 a(2^n) = n.
%e A327503 The transformation A327502 takes 144 -> 2 -> 1, so a(144) = 2.
%e A327503 From _Antti Karttunen_, Apr 03 2022: (Start)
%e A327503 For n = 1728 = 2^6 * 3^3, A327501(1728) = 864 = 2^5 * 3^3, and therefore A327502(1728) = 1728/864 = 2. A327501(2) = 2, thus A327502(2) = 2/2 = 1, so we reached 1 (= A327502(1)) in two steps, and therefore a(1728) = 2.
%e A327503 For n = 125000 = 2^3 * 5^6, A327501(125000) = 31250 = 2^1 * 5^6, and therefore A327502(125000) = 125000/31250 = 4. A327501(4) = 2, thus A327502(4) = 4/2 = 2, from which we reach 1 in one more step, therefore a(125000) = 3.
%e A327503 (End)
%t A327503 Table[Length[FixedPointList[#/Max[Select[Divisors[#],GCD@@Last/@FactorInteger[#]==1&]]&,n]]-2,{n,100}]
%o A327503 (PARI)
%o A327503 A327502(n) = if(n==1, 1, n/vecmax(select(x->((x>1) && !ispower(x)), divisors(n))));
%o A327503 A327503(n) = { my(u=A327502(n)); if(u==n, 0, 1+A327503(u)); }; \\ _Antti Karttunen_, Apr 02 2022
%Y A327503 See link for additional cross-references.
%Y A327503 Cf. A000961, A001597, A007916 (positions of 1's), A052409, A158378, A303386, A327501, A327502, A351948.
%Y A327503 Cf. also A327500.
%K A327503 nonn
%O A327503 1,4
%A A327503 _Gus Wiseman_, Sep 16 2019
%E A327503 Data section extended up to 105 terms by _Antti Karttunen_, Apr 02 2022