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

273, 63973, 72719023, 13006678091, 7817013532691

Offset

3,1

Comments

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.

Entries verified computationally.

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

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

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

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

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

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

Links

Table of n, a(n) for n=3..7.

B. Fagin, Idempotent Factorizations of Square-Free Integers, Information 2019, 10(7), 232.

Example

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.

Mathematica

(* This program is not suitable to compute large terms. *)
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]&]];
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 *)

Prog

(Python)
# Partial Python code is shown below.  It uses other routines:
# numbthy.factor(n) -- from the Python number theory library, returns a list of
# (p, e) pairs corresponding to the prime factors and their exponents in the factorizations of n
# partitions(n, factor_list) -- takes an integer n and the factor list from above,
# returns a list of all bipartite factorizations of n
# lambda_n -- calculates the carmichael lambda function
# returns True if all partitions of n are idempotent
def isMaximallyIdempotent(n):
    factor_list = numbthy.factor(n)
    partitions_of_n = partitions(n, factor_list)
    lambda_n = carmichael_lambda_with_list(n, factor_list)
    for (p, q) in partitions_of_n:
        pseudo = (p-1)*(q-1)
        if pseudo % lambda_n != 0:
            return False
    return True

Crossrefs

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

Sequence in context: A351766 A321035 A225702 * A295455 A174771 A031417

Adjacent sequences: A307534 A307535 A307536 * A307538 A307539 A307540

Keyword

nonn,more

Author

Barry Fagin, Apr 13 2019

Extensions

M(7), now confirmed as being a(7), added by Barry Fagin, Dec 04 2019

Status

approved