A308550 - OEIS

%I #11 Jun 22 2019 14:38:19

%S 2,7,11,14,19,25,33

%N a(n) is the largest number k such that all subset sums of the n-th powers of the first k primes are distinct.

%e a(1)=2 because the subsets of {prime(1)^1, prime(2)^1} = {2^1, 3^1} = {2, 3} are {}, {2}, {3}, and {2, 3}, whose sums are 0, 2, 3, and 5 (all distinct), but the subsets of {prime(1)^1, prime(2)^1, prime(3)^1} = {2, 3, 5} include {2,3} and {5}, whose sums are both 5.

%e a(2)=7 because the sums of subsets of {prime(1)^2, ..., prime(7)^2} are all distinct, but the subsets of {prime(1)^2, ..., prime(8)^2} include {11^2, 17^2} and {7^2, 19^2}, whose sums are both 410.

%e a(3)=11 because the sums of subsets of {2^3, 3^3, 5^3, ..., prime(11)^3} are all distinct, but the subsets of {2^3, 3^3, 5^3, ..., prime(12)^3} include {2^3, 7^3, 11^3, 13^3, 17^3, 29^3, 31^3} and {3^3, 5^3, 23^3, 37^3}, whose sums are both 62972.

%e a(4)=14; the subsets of {2^4, 3^4, 5^4, ..., prime(15)^4} include {11^4, 23^4, 41^4, 43^4} and {13^4, 29^4, 31^4, 47^4}, whose sums are both 6539044.

%e a(5)=19; 23^5 + 29^5 + 31^5 + 61^5 + 67^5 = 13^5 + 37^5 + 43^5 + 47^5 + 71^5 = 2250298051.

%e From _Bert Dobbelaere_, Jun 22 2019: (Start)

%e a(6)=25; 13^6 + 31^6 + 37^6 + 41^6 + 43^6 + 53^6 + 61^6 + 73^6 + 97^6 = 2^6 + 3^6 + 5^6 + 7^6 + 11^6 + 17^6 + 19^6 + 23^6 + 47^6 + 101^6.

%e a(7)=33; 3^7 + 5^7 + 7^7 + 11^7 + 17^7 + 37^7 + 41^7 + 83^7 + 89^7 + 101^7 + 131^7 + 137^7 = 2^7 + 13^7 + 29^7 + 31^7 + 43^7 + 53^7 + 67^7 + 97^7 + 103^7 + 127^7 + 139^7. (End)

%K nonn,more

%O 1,1

%A _Jon E. Schoenfield_, Jun 06 2019

%E a(6)-a(7) from _Bert Dobbelaere_, Jun 22 2019