cp's OEIS Frontend

A332066 Number of positive integers whose n-th power is not the sum of distinct smaller positive n-th powers.

%I A332066 #25 Jun 09 2021 02:29:23
%S A332066 2,6,9,32,24,30,41,83,49,62,71,83
%N A332066 Number of positive integers whose n-th power is not the sum of distinct smaller positive n-th powers.
%C A332066 See A332065 for the numbers whose n-th power is the sum of distinct smaller positive n-th powers. This sequence counts the positive integers not in a given row n of that table, whence the formula.
%F A332066 a(n) = lim_{k -> oo} A332065(n,k) - k.
%F A332066 a(n) <= A332098(n) with equality iff A030052(n) = A332098(n) + 1 <=> A030052(n) > A332098(n), which happens for n = 1, 8, 10, ... The difference A332098(n) - a(n) is the number of "solutions" s (listed in rows of A332065) strictly less than the largest "non-solution" A332098(n).
%e A332066 For n = 1, only s = 1 and s = 2 are not the sum of distinct smaller positive integers (to the power n = 1), for all s >= 3 on we have s^1 = 1^1 + (s-1)^1 with 1 and s-1 distinct positive integers. Thus a(1) = #{1, 2} = 2.
%e A332066 For n = 2, S2 = {1, 2, 3, 4, 6, 8} is the set of all s > 0 whose square is not the sum of distinct smaller squares, while 5^2 = 4^2 + 3^2, 7^2 = 6^2 + 3^2 + 2^2, and all s^2 >= 9^2 are also the sum of distinct smaller squares. Thus a(2) = #S2 = 6.
%Y A332066 Cf. A332065, A030052, A332096, A332097, A332098, A332066.
%K A332066 nonn,more
%O A332066 1,1
%A A332066 _M. F. Hasler_, Jul 19 2020