We present an analysis of a 37-ks observation of the supergiant fast X-ray transient (SFXT) IGR J17391-3021 (=XTE J1739-302) gathered with Suzaku. The source evolved from quiescence to a low-activity level culminating in three weak flares lasting ~3 ks each in which the peak luminosity is only a factor of 5 times that of the pre-flare luminosity. The minimum observed luminosity was 1.3x10^33 erg/s (d/2.7 kpc)^2 in the 0.5--10 keV range. The weak flares are accompanied by significant changes in the spectral parameters including a column density (nH = (4.1+-0.5)x10^22 /cm^2) that is ~2--9 times the absorption measured during quiescence. Accretion of obscuring clumps of stellar wind material can explain both the small flares and the increase in nH. Placing this observation in the context of the recent Swift monitoring campaign, we find that weak-flaring episodes, or at least epochs of enhanced activity just above the quiescent level but well below the moderately bright or high-luminosity outbursts, represent more than 60+-5% of all observations in the 0.5--10keV energy range making this the most common state in the emission behavior of IGR J17391-3021
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