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 A380218 #13 Jan 28 2025 20:51:42
%S A380218 1,0,0,0,0,1,0,2,1,1,0,3,0,1,1,4,0,3,0,3,1,1,0,6,1,1,2,3,0,4,0,6,1,1,
%T A380218 1,8,0,1,1,6,0,4,0,3,3,1,0,11,1,3,1,3,0,6,1,6,1,1,0,10,0,1,3,10,1,4,0,
%U A380218 3,1,4,0,15,0,1,3,3,1,4,0,11,4,1,0,10,1,1,1,6,0,10,1,3,1,1,1,18,0,3,3,8,0,4,0,6,4,1
%N A380218 Number of integer partitions of n with product n+1.
%C A380218 For n instead of n+1 we have 0 followed by A001055.
%C A380218 Also the number of integer factorizations of n with sum < n. [When the sequence is interpreted as having offset 1 instead of 0. Clarified by _Antti Karttunen_, Jan 28 2025]
%H A380218 Antti Karttunen, <a href="/A380218/b380218.txt">Table of n, a(n) for n = 0..20000</a>
%e A380218 The a(5) = 1 through a(15) = 4 partitions with product n+1:
%e A380218 32 . 421 3311 5211 . 62111 . 721111 53111111 8211111
%e A380218 2221 431111 441111111
%e A380218 3221111 4221111111
%e A380218 22221111111
%e A380218 The a(1) = 1 through a(12) = 3 factorizations with sum < n:
%e A380218 () . . . . (2*3) . (2*4) (3*3) (2*5) . (2*6)
%e A380218 (2*2*2) (3*4)
%e A380218 (2*2*3)
%t A380218 Table[Length[Select[IntegerPartitions[n],SameQ[Times@@#,n+1]&]],{n,0,30}]
%o A380218 (PARI)
%o A380218 A380218off1(n, m=n, e=n) = if(1==n, (e>0), sumdiv(n, d, if((d>1)&&(d<=m), A380218off1(n/d, d, e-d))));
%o A380218 A380218off0(n) = A380218off1(1+n); \\ _Antti Karttunen_, Jan 28 2025
%Y A380218 Same as A028422 = A001055-1 except initial terms.
%Y A380218 These partitions are ranked by A325041.
%Y A380218 The version for divisibility instead of equality is A379320.
%Y A380218 A000041 counts integer partitions, strict A000009.
%Y A380218 A379666 counts partitions by sum and product.
%Y A380218 A380219 counts partitions of n whose product is a proper multiple of n, ranks A380216.
%Y A380218 Counting and ranking multisets by comparing sum and product:
%Y A380218 - same: A001055, ranks A301987
%Y A380218 - multiple: A057567, ranks A326155
%Y A380218 - divisor: A057568 (strict A379733), ranks A326149, see A379319, A380217.
%Y A380218 - greater than: A096276 shifted right, ranks A325038
%Y A380218 - greater or equal: A096276, ranks A325044
%Y A380218 - less than: A114324, ranks A325037, see A318029, A379720
%Y A380218 - less or equal: A319005, ranks A379721, see A025147
%Y A380218 - different: A379736, ranks A379722, see A111133
%Y A380218 Cf. A003963, A069016, A318950, A319000, A319916, A325042, A326152, A326156, A379734.
%K A380218 nonn
%O A380218 0,8
%A A380218 _Gus Wiseman_, Jan 21 2025
%E A380218 More terms from _Antti Karttunen_, Jan 28 2025