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Each term is the sum of the seventh powers of three or more of its prime factors (since the sum of seventh powers of two distinct primes would not be divisible by those primes).
It is possible that the three terms shown are just the smallest examples presently known - there may be smaller ones.
Other terms include the following (and these too may not be the next terms):
48174957112005843444270083236899591347874 = 2^7 + 1259^7 + 648383^7.
343628633008268493930426179988576850614546787655 = 5^7 + 97^7 + 6178313^7.
1556588247952374145751498792380776025975963817566087335 = 5^7 + 941^7 + 55174589^7.
6777869034345885139001456808449377853222864558972446987604 = 2^7 + 337^7 + 182635307^7.
8652931112104420195217156139788964690213217995925746635175635 = 5^7 + 29^7 + 507351601^7.
33684756195335243623428442147352712728560450053586233129585039130540009686445977 = 3^7 + 2731^7 + 229647602339^7.
4218418507660286246537768294375414778864666339784229288571328866079146694717894140 = 5^7 + 7^7 + 2677^7 + 457863123059^7.
(End)
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