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 A373271 #15 Jun 02 2024 19:32:45
%S A373271 1,2,3,6,10,14,24,34,49,70,103,134,195,258,347,461,624,796,1066,1358,
%T A373271 1763,2250,2903,3631,4644,5805,7309,9083,11381,13998,17428,21369,
%U A373271 26336,32174,39451,47847,58399,70610,85590,103077,124462,149169,179368,214300,256397
%N A373271 a(n) = sum for all integer partitions of n of the number of distinct multiplicities in each partition.
%C A373271 Sum of the rows of A373269.
%C A373271 Sum of the rows of A373270.
%C A373271 The multiplicity of a part in an integer partition (or composition) is the number of times it appears in the partition, seen as a list.
%C A373271 The multiplicity of 3 in the partition 12 = 5+3+3+1 is 2.
%C A373271 For this sequence, only distinct multiplicities appearing for parts of the partition are counted, only once for a given partition.
%C A373271 If all multiplicities of all parts of all integer partitions of n are counted, one gets A000070 (1, 2, 4, 7, 12, 19, 30, 45, 67, 97, ...).
%C A373271 If all distinct multiplicities of all parts of all integer partitions are summed, one gets A373273 (1, 3, 5, 11, 18, 29, 48, 74, 107, 161, ...).
%C A373271 If all multiplicities of all parts of all integer partitions of n are summed, one gets A006128 (1, 3, 6, 12, 20, 35, 54, 86, 128, 192, ...).
%H A373271 Alois P. Heinz, <a href="/A373271/b373271.txt">Table of n, a(n) for n = 1..200</a>
%e A373271 Example for n=20:
%e A373271 the partition 20=4+3+3+3+3+2+1+1
%e A373271 has multiplicities 1, 4, 1, 2
%e A373271 for the parts 4,3,2,1 listed in descending order.
%e A373271 It has 3 different multiplicities (1, 2 and 4) and contributes 3 to a(20) = 1358.
%t A373271 Table[Plus @@
%t A373271 Table[Plus @@
%t A373271 Map[Length[Union[Length /@ Split[#]]] &,
%t A373271 IntegerPartitions[n, {k}]], {k, 1, n}], {n, 1, 40}]
%Y A373271 Cf. A373243, A098859.
%K A373271 nonn
%O A373271 1,2
%A A373271 _Olivier Gérard_, May 29 2024