This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A383683 #10 May 06 2025 15:20:40
%S A383683 1,1,2,3,5,7,11,15,21,29,39,52,68,89,117,150,192,244,309,387,485,603,
%T A383683 749,922,1130,1384,1680,2035,2440,2922,3478,4118,4867,5728,6740,7879,
%U A383683 9206,10741,12502,14516,16846,19533,22620,26164,30252,34967,40450,46786
%N A383683 The number of possible values that can be obtained for the Shannon diversity index across all partitions of n.
%C A383683 For a partition P of n into parts (n_1, n_2, ..., n_k), the Shannon diversity index is S(P) = -Sum_{i=1..k} (n_i/n)*log(n_i/n). a(n) is the number of distinct values that S(P) obtains across all possible partitions P of n.
%e A383683 For n=0 through 7, each partition of n produces a distinct value of the Shannon diversity index, so that a(n) is equal to the number of partitions, A000041(n).
%e A383683 For n=8, partitions (2,2,2,2) and (4,1,1,1,1) both have the same Shannon diversity index, 2*log(2), so that a(8) = 21, one less than A000041(8).
%Y A383683 A000607 provides a lower bound for a(n).
%Y A383683 Cf. A000041.
%K A383683 nonn
%O A383683 0,3
%A A383683 _Noah A Rosenberg_, May 05 2025