A307537 - OEIS

%I #38 Dec 30 2019 12:35:16

%S 273,63973,72719023,13006678091,7817013532691

%N a(n) is the smallest maximally idempotent integer with n factors, n >= 3.

%C Maximally idempotent integers are those squarefree integers such that all their bipartite factorizations are idempotent (see A306812). All squarefree integers with n <= 2 factors have this property, and are therefore excluded from the definition.

%C Entries verified computationally.

%C The lambda values and factorizations of the integers in this sequence are:

%C   M(3) = 3*7*13, lambda = 12;

%C   M(4) = 7*13*19*37, lambda = 36;

%C   M(5) = 13*19*37*73*109, lambda = 216;

%C   M(6) = 11*31*41*61*101*151, lambda = 600;

%C   M(7) = 11*31*41*61*101*151*601, lambda = 600.

%H B. Fagin, <a href="https://doi.org/10.3390/info10070232"> Idempotent Factorizations of Square-Free Integers</a>, Information 2019, 10(7), 232.

%e 273 is the smallest maximally idempotent integer. Factorization is (3,7,13). Bipartite factorizations are (3,91), (7,39), (13,21). Lambda(273) = 12, (2*90),(6*38) and (12*20) are all divisible by 12, thus 273 is maximally idempotent.

%t (* This program is not suitable to compute large terms. *)

%t okQ[n_] := Module[{partitions, p, q, lambda}, partitions = {p, q} /. {ToRules[Reduce[1<p<q && n == p q, {p, q}, Integers]]}; lambda = CarmichaelLambda[n]; AllTrue[partitions-1, Divisible[Times @@ #, lambda]&]];

%t For[Clear[a]; n = 1, n < 70000, n++, If[SquareFreeQ[n], nu = PrimeNu[n]; If[nu >= 3 && !IntegerQ[a[nu]], If[okQ[n], Print["a(", nu, ") = ", n]; a[nu] = n]]]]; (* _Jean-François Alcover_, Jun 20 2019 *)

%o (Python)

%o # Partial Python code is shown below.  It uses other routines:

%o # numbthy.factor(n) -- from the Python number theory library, returns a list of

%o # (p,e) pairs corresponding to the prime factors and their exponents in the factorizations of n

%o # partitions(n,factor_list) -- takes an integer n and the factor list from above,

%o # returns a list of all bipartite factorizations of n

%o # lambda_n -- calculates the carmichael lambda function

%o # returns True if all partitions of n are idempotent

%o def isMaximallyIdempotent(n):

%o     factor_list = numbthy.factor(n)

%o     partitions_of_n = partitions(n,factor_list)

%o     lambda_n = carmichael_lambda_with_list(n,factor_list)

%o     for (p,q) in partitions_of_n:

%o         pseudo = (p-1)*(q-1)

%o         if pseudo % lambda_n != 0:

%o             return False

%o     return True

%Y Cf. A005117, A120944, A306330, A306508, A306812.

%K nonn,more

%O 3,1

%A _Barry Fagin_, Apr 13 2019

%E M(7), now confirmed as being a(7), added by _Barry Fagin_, Dec 04 2019