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A307537 a(n) is the smallest maximally idempotent integer with n factors, n >= 3.

%I A307537 #38 Dec 30 2019 12:35:16
%S A307537 273,63973,72719023,13006678091,7817013532691
%N A307537 a(n) is the smallest maximally idempotent integer with n factors, n >= 3.
%C A307537 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 A307537 Entries verified computationally.
%C A307537 The lambda values and factorizations of the integers in this sequence are:
%C A307537   M(3) = 3*7*13, lambda = 12;
%C A307537   M(4) = 7*13*19*37, lambda = 36;
%C A307537   M(5) = 13*19*37*73*109, lambda = 216;
%C A307537   M(6) = 11*31*41*61*101*151, lambda = 600;
%C A307537   M(7) = 11*31*41*61*101*151*601, lambda = 600.
%H A307537 B. Fagin, <a href="https://doi.org/10.3390/info10070232"> Idempotent Factorizations of Square-Free Integers</a>, Information 2019, 10(7), 232.
%e A307537 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 A307537 (* This program is not suitable to compute large terms. *)
%t A307537 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 A307537 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 A307537 (Python)
%o A307537 # Partial Python code is shown below.  It uses other routines:
%o A307537 # numbthy.factor(n) -- from the Python number theory library, returns a list of
%o A307537 # (p,e) pairs corresponding to the prime factors and their exponents in the factorizations of n
%o A307537 # partitions(n,factor_list) -- takes an integer n and the factor list from above,
%o A307537 # returns a list of all bipartite factorizations of n
%o A307537 # lambda_n -- calculates the carmichael lambda function
%o A307537 # returns True if all partitions of n are idempotent
%o A307537 def isMaximallyIdempotent(n):
%o A307537     factor_list = numbthy.factor(n)
%o A307537     partitions_of_n = partitions(n,factor_list)
%o A307537     lambda_n = carmichael_lambda_with_list(n,factor_list)
%o A307537     for (p,q) in partitions_of_n:
%o A307537         pseudo = (p-1)*(q-1)
%o A307537         if pseudo % lambda_n != 0:
%o A307537             return False
%o A307537     return True
%Y A307537 Cf. A005117, A120944, A306330, A306508, A306812.
%K A307537 nonn,more
%O A307537 3,1
%A A307537 _Barry Fagin_, Apr 13 2019
%E A307537 M(7), now confirmed as being a(7), added by _Barry Fagin_, Dec 04 2019