A257659 - OEIS

%I #29 Nov 03 2016 18:35:03

%S 275223438741,4561072096211306682,9306119954843409393442022085025276

%N Numbers that are not seventh powers, but can be written as the sum of the seventh powers of two or more of their prime factors.

%C From _Robert Israel_, Nov 02 2016: (Start)

%C 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).

%C It is possible that the three terms shown are just the smallest examples presently known - there may be smaller ones.

%C Other terms include the following (and these too may not be the next terms):

%C 48174957112005843444270083236899591347874 = 2^7 + 1259^7 + 648383^7.

%C 343628633008268493930426179988576850614546787655 = 5^7 + 97^7 + 6178313^7.

%C 1556588247952374145751498792380776025975963817566087335 = 5^7 + 941^7 + 55174589^7.

%C 6777869034345885139001456808449377853222864558972446987604 = 2^7 + 337^7 + 182635307^7.

%C 8652931112104420195217156139788964690213217995925746635175635 = 5^7 + 29^7 + 507351601^7.

%C 33684756195335243623428442147352712728560450053586233129585039130540009686445977 = 3^7 + 2731^7 + 229647602339^7.

%C 4218418507660286246537768294375414778864666339784229288571328866079146694717894140 = 5^7 + 7^7 + 2677^7 + 457863123059^7.

%C (End)

%D J. M. De Koninck, Those Fascinating Numbers, American Mathematical Society, 2009, page 362, ISBN 978-0-8218-4807-4.

%H Jean-Marie De Koninck & Florian Luca, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Koninck/koninck71.html">Partial Sums of Powers of Prime Factors</a>, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.6 (see p. 7).

%e 275223438741 is not a seventh power, i.e., not a term of A001015, but is equal to the product of prime numbers 3 * 23 * 43 * 92761523, and 3^7 + 23^7 + 43^7 = 275223438741, so 275223438741 is a term of the sequence.

%Y Cf. A001015, A092759.

%K nonn,more

%O 1,1

%A _Felix Fröhlich_, Jul 26 2015

%E Edited by _Robert Israel_, Nov 02 2016