According to Bard the Google AI engine:
Bhagavata Saptaham is a seven-day religious discourse on the Srimad Bhagavatam, one of the eighteen major Hindu Puranas. It is a popular tradition in Hinduism, and is often held in temples, ashrams, and other religious institutions.
Does any scripture mention the significance of the discourse lasting exactly seven days?
This is explained in a section of the Padma Purana, the Srimad Bhagavata Mahatyam.
(There isn't a neat, single paragraph to quote, so I'm going to quote fairly extensively. There is still a lot more context you need, so I suggest reading the whole linked translation if you're confused.)
How those devotees live in this world without you being here ? It is difficult to worship the One without any form. Therefore, be concerned with this at least a little.
Upon listening to these words from Uddhava at the sacred place of Prabhasa, Bhagavan started thinking: " For the purpose of the protection of the devotees what I am supposed to do?"
Bhagavan assimilated all his powers into the Srimad Bhagavata Mahapurana. He disappeared from here and entered into the great ocean of Srimad Bhagavatam.
Therefore, Srimad Bhagavaram is the embodiment of Sri Krishna in the form of words in real terms. Through serving this great Mahapurana, listening to it, reading it and looking at it annihilates all the sins in the human beings.
Because of this reason, listening to Srimad Bhagavatam in full within seven days has been determined as the most greatest. In this time of Kali, discarding all other methods, following this path alone has been considered as the most appropriate.
Sorrows, poverty, bad luck, sins etc. get washed out. Wins over the lust and anger. This (following Srimad Bhagavatam listening, reading and remembering) has been prescribed as the most suitable one in the times of Kali.
I don't see the logic here ('For this reason'... For what reason?), but there is a story related early in the section that tells the significance.
It comes at the end of the story of Gokarna and Dundukhari, which you can read about here.
From different kingdoms and villages people reached there with the intention of listening to Srimad Bhagavatam. Lame persons, blind, old people, dull-headed - like of them all reached continuously in order to get rid of the effect of their bad deeds.
Even the Devas got surprised as to how so many people got assembled there. When Gokarna occupied the centre stage (Vyasa Peetam) and started the discourse, the ghost also reached there. When he started searching here and there for himself to sit, he spotted a bamboo tree planted straight which had seven knot points, each one of them having one hole each.
That ghost entered through the hole at the bottom of the bamboo and placed itself there for listening to the discourse. As the ghost was in the form of air, it was not in a position to station itself at one place. Therefore, it placed itself within that compartment of the bamboo.
As far as Gokarna, the son of the cow, was concerned, having made a Vaishnava Brahmin seated on the front as the prime listener, started the discourse of Srimad Bhagavatam from the first volume.
As the evening approached, he stopped the discourse. At that time there occurred a wonderful development. As the people over there watched, making a big noise one compartment of the bamboo split open.
At the end of the evening of the second day, the second compartment of the bamboo split in the same manner. Again, at the end of the evening of the third day, the third one also split.
Within these seven days all the seven compartments of the bamboo got split in the even manner. After listening to the 12th Volume of Srimad Bhagavatam, that Dhundhukari could discard his ghost form and thereby got liberated.
In the story, Gokarna's brother Dundukhari had been trapped in the form of a ghost. When Gokarna recited the Bhagavatam, each day his brother's spirit was freed a little bit more, until on the seventh day, his brother finally broke free. Many of the other impaired people there were also cured upon the recitation's ending. This would have established that the Bhagavata Saptaham was an effective form of recitation.
One last quote, stanzas 71-73:
Sri Sutha said (to the Saunakas):
After explaining all these, those great men (the Sanaka Saints), delivered the discourse of Srimad Bhagavatam which is so pious and which ensures deliverance from all the worldly miseries, in seven days in the prescribed manner (Sapthaham) when all the living beings there listened to them with rapt attention and concentrated mind. After the conclusion of the discourse, they recited praises in the name of Sri Bhagavan.After all these happened, the Bhakti, Jnana and Vairagya got nourished appropriately. They also became so attractive and youthful to the minds of all the people.
Sorry that this is so long!
