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 A379720 #12 Jan 12 2025 19:10:54
%S A379720 1,0,0,0,0,1,3,3,6,7,11,13,20,23,33,40,54,65,87,104,136,164,209,252,
%T A379720 319,382,477,573,707,846,1038,1237,1506,1793,2166,2572,3093,3659,4377,
%U A379720 5169,6152,7244,8590,10086,11913,13958,16423,19195,22518,26251,30700,35716
%N A379720 Except a(0)=1 and a(4)=0, number of integer partitions of n with no 1's and at least two parts.
%C A379720 Also partitions of n such that all parts are > 1 and have product > n.
%C A379720 Allowing 1's gives A114324, ranks A325037. The strict case is A318029 (except first term).
%F A379720 Except for n = 0 and n = 4, we have a(n) = A002865(n) - 1.
%e A379720 The a(5) = 1 through a(11) = 13 partitions:
%e A379720 (3,2) (3,3) (4,3) (4,4) (5,4) (5,5) (6,5)
%e A379720 (4,2) (5,2) (5,3) (6,3) (6,4) (7,4)
%e A379720 (2,2,2) (3,2,2) (6,2) (7,2) (7,3) (8,3)
%e A379720 (3,3,2) (3,3,3) (8,2) (9,2)
%e A379720 (4,2,2) (4,3,2) (4,3,3) (4,4,3)
%e A379720 (2,2,2,2) (5,2,2) (4,4,2) (5,3,3)
%e A379720 (3,2,2,2) (5,3,2) (5,4,2)
%e A379720 (6,2,2) (6,3,2)
%e A379720 (3,3,2,2) (7,2,2)
%e A379720 (4,2,2,2) (3,3,3,2)
%e A379720 (2,2,2,2,2) (4,3,2,2)
%e A379720 (5,2,2,2)
%e A379720 (3,2,2,2,2)
%t A379720 Table[Length[Select[IntegerPartitions[n],FreeQ[#,1]&&Plus@@#<Times@@#&]],{n,0,30}]
%Y A379720 For <= instead of < we have A002865 = partitions into parts > 1.
%Y A379720 These partitions have ranks A071904 (except initial terms).
%Y A379720 Set a(4) = 1 to get A083751.
%Y A379720 A000041 counts integer partitions, strict A000009.
%Y A379720 A379668 counts partitions without 1's by sum and product.
%Y A379720 Counting and ranking multisets by comparing sum and product:
%Y A379720 - same: A001055, ranks A301987
%Y A379720 - divisible: A057567, ranks A326155
%Y A379720 - divisor: A057568, ranks A326149, see A379733
%Y A379720 - greater than: A096276 shifted right, ranks A325038
%Y A379720 - greater or equal: A096276, ranks A325044
%Y A379720 - less than: A114324, ranks A325037, see A318029
%Y A379720 - less or equal: A319005, ranks A379721, see A025147
%Y A379720 - different: A379736, ranks A379722, see A111133
%Y A379720 Cf. A003963, A028422, A318950, A319000, A319916, A325036, A325041, A326152, A326178, A379666, A379678.
%K A379720 nonn
%O A379720 0,7
%A A379720 _Gus Wiseman_, Jan 06 2025