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%I #23 Jul 20 2019 11:04:13
%S 5,103,59,127,151,547,307
%N Smallest prime whose n-th power can be represented as the sum of the n-th powers of two or more distinct primes; or -1 if no such prime exists.
%C From _Bert Dobbelaere_, Jul 19 2019: (Start)
%C a(8) > 800 or a(8) = -1.
%C Any sum of 6th powers of distinct primes that adds up to a 6th power of a prime must have at least 65 terms (see expression for a(6)^6 in example). This follows from the fact that for any prime p except 2, 3 and 7, p^6 == 1 (mod 504). A similar argument using modulus 480 gives us a lower bound of 97 terms for a(8)^8.
%C (End)
%H dxdy forum, <a href="https://dxdy.ru/post1395207.html">post</a> (in Russian).
%e a(1) = 5, because 5^1 = 2^1 + 3^1.
%e a(2) = 103, because 103^2 = 2^2 + 3^2 + 5^2 + 7^2 + 11^2 + 13^2 + 17^2 + 19^2 + 23^2 + 29^2 + 31^2 + 41^2 + 43^2 + 61^2.
%e a(3) = 59, because 59^3 = 5^3 + 13^3 + 29^3 + 31^3 + 53^3.
%e a(4) = 127, because 127^4 = 3^4 + 5^4 + 7^4 + 11^4 + 13^4 + 17^4 + 23^4 + 29^4 + 31^4 + 43^4 + 47^4 + 59^4 + 61^4 + 67^4 + 73^4 + 89^4 + 103^4.
%e a(5) = 151, because 151^5 = 2^5 + 5^5 + 7^5 + 11^5 + 13^5 + 17^5 + 31^5 + 37^5 + 43^5 + 47^5 + 53^5 + 59^5 + 71^5 + 73^5 + 79^5 + 101^5 + 103^5 + 107^5 + 109^5 + 113^5.
%e a(6) = 547, because 547^6 = 3^6 + 5^6 + 7^6 + 11^6 + 17^6 + 19^6 + 23^6 + 29^6 + 31^6 + 41^6 + 43^6 + 47^6 + 53^6 + 59^6 + 67^6 + 71^6 + 73^6 + 79^6 + 83^6 + 97^6 + 101^6 + 103^6 + 109^6 + 113^6 + 127^6 + 131^6 + 137^6 + 139^6 + 149^6 + 151^6 + 157^6 + 163^6 + 167^6 + 173^6 + 179^6 + 181^6 + 191^6 + 193^6 + 197^6 + 199^6 + 211^6 + 223^6 + 227^6 + 229^6 + 233^6 + 239^6 + 251^6 + 257^6 + 263^6 + 269^6 + 271^6 + 277^6 + 281^6 + 293^6 + 307^6 + 311^6 + 313^6 + 317^6 + 337^6 + 347^6 + 353^6 + 359^6 + 367^6 + 389^6 + 419^6. - _Bert Dobbelaere_, Jul 18 2019
%e a(7) = 307, because 307^7 = 11^7 + 13^7 + 23^7 + 29^7 + 31^7 + 41^7 + 43^7 + 47^7 + 59^7 + 73^7 + 89^7 + 97^7 + 103^7 + 109^7 + 113^7 + 127^7 + 131^7 + 137^7 + 139^7 + 149^7 + 151^7 + 167^7 + 191^7 + 193^7 + 197^7 + 199^7 + 223^7 + 227^7 + 233^7 + 239^7 + 251^7. - _Bert Dobbelaere_, Jul 17 2019
%e Note that these are the smallest prime powers for which such a representation is possible.
%Y Cf. A308357.
%K nonn,more,hard
%O 1,1
%A _Dmitry Kamenetsky_, May 28 2019
%E a(6)-a(7) from _Bert Dobbelaere_, Jul 18 2019