A307537 a(n) is the smallest maximally idempotent integer with n factors, n >= 3.
273, 63973, 72719023, 13006678091, 7817013532691
Offset: 3
Examples
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.
Links
- B. Fagin, Idempotent Factorizations of Square-Free Integers, Information 2019, 10(7), 232.
Programs
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Mathematica
(* This program is not suitable to compute large terms. *) okQ[n_] := Module[{partitions, p, q, lambda}, partitions = {p, q} /. {ToRules[Reduce[1 -
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
Extensions
M(7), now confirmed as being a(7), added by Barry Fagin, Dec 04 2019
Comments