A351065 - OEIS

%I #20 Mar 02 2022 12:46:22

%S 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,1,1,1,1,2,1,1,1,1,1,1,1,1,2,2,

%T 2,1,1,1,1,1,2,2,2,1,1,1,1,2,1,1,1,2,2,2,2,2,2,2,1,2,2,2,1,3,4,4,4,3,

%U 3,3,3,2,2,2,2,1,1,1,1,5,2,2,2,4,1,1,1,3,4,1,2,3,1,1,2,3,2,3,3,1,1,1,1,2,6,1,4

%N Number of different ways to obtain n as a sum of the minimal possible number of positive perfect powers with different exponents (considering only minimal possible exponents for bases equal to 1).

%C Every positive integer k appears in the sequence, as a(2^(2^k)) = k.

%e a(4) = 1, because 4 = 2^2 is its only possible representation, and similarly for every power a^p, with a > 1 and p prime.

%e a(16) = 2, because 16 = 2^4 = 4^2. More generally, a^(p^2) -- with a > 1 and p prime -- can be written in exactly two ways.

%e a(17) = 3, because 17 = 1^2 + 2^4 = 3^2 + 2^3 = 4^2 + 1^3.

%e a(313) = 10, because 313 can be written in exactly 10 different ways (with three perfect powers): 4^2 + 6^3 + 3^4 = 5^2 + 2^5 + 2^8 = 5^2 + 4^4 + 2^5 = 7^2 + 2^3 + 2^8 = 7^2 + 2^3 + 4^4 = 9^2 + 6^3 + 2^4 = 11^2 + 2^6 + 2^7 = 11^2 + 4^3 + 2^7 = 13^2 + 2^4 + 2^7 = 17^2 + 2^3 + 2^4.

%Y Cf. A351062, A351063, A351064, A351066.

%K nonn

%O 1,16

%A _Alberto Zanoni_, Feb 22 2022