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).

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, 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, 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

Offset

1,4

Comments

First differs from A052409 and A158378 at a(216) = 2, A052409(216) = A158378(216) = 3.

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).

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

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Gus Wiseman, Sequences counting and encoding certain classes of multisets

Formula

a(2^n) = n.

Example

The transformation A327502 takes 144 -> 2 -> 1, so a(144) = 2.
From Antti Karttunen, Apr 03 2022: (Start)
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.
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.
(End)

Mathematica

Table[Length[FixedPointList[#/Max[Select[Divisors[#], GCD@@Last/@FactorInteger[#]==1&]]&, n]]-2, {n, 100}]

Prog

(PARI)
A327502(n) = if(n==1, 1, n/vecmax(select(x->((x>1) && !ispower(x)), divisors(n))));
A327503(n) = { my(u=A327502(n)); if(u==n, 0, 1+A327503(u)); }; \\ Antti Karttunen, Apr 02 2022

Crossrefs

See link for additional cross-references.

Cf. A000961, A001597, A007916 (positions of 1's), A052409, A158378, A303386, A327501, A327502, A351948.

Cf. also A327500.

Sequence in context: A253641 A158378 A052409 * A051904 A070012 A375034

Adjacent sequences: A327500 A327501 A327502 * A327504 A327505 A327506

Keyword

nonn

Author

Gus Wiseman, Sep 16 2019

Extensions

Data section extended up to 105 terms by Antti Karttunen, Apr 02 2022

Status

approved