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 A371178 #19 Mar 18 2024 11:45:47
%S A371178 1,1,1,2,3,4,6,9,12,16,21,28,37,48,62,80,101,127,162,202,252,312,386,
%T A371178 475,585,713,869,1056,1278,1541,1859,2232,2675,3196,3811,4534,5386,
%U A371178 6379,7547,8908,10497,12345,14501,16999,19897,23253,27135,31618,36796,42756
%N A371178 Number of integer partitions of n containing all divisors of all parts.
%C A371178 The Heinz numbers of these partitions are given by A371177.
%C A371178 Also partitions such that the number of distinct parts is equal to the number of distinct divisors of parts.
%e A371178 The partition (4,2,1,1) contains all distinct divisors {1,2,4}, so is counted under a(8).
%e A371178 The partition (4,4,3,2,2,2,1) contains all distinct divisors {1,2,3,4} so is counted under 4 + 4 + 3 + 2 + 2 + 2 + 1 = 18. - _David A. Corneth_, Mar 18 2024
%e A371178 The a(0) = 1 through a(8) = 12 partitions:
%e A371178 () (1) (11) (21) (31) (221) (51) (331) (71)
%e A371178 (111) (211) (311) (321) (421) (521)
%e A371178 (1111) (2111) (2211) (511) (3221)
%e A371178 (11111) (3111) (2221) (3311)
%e A371178 (21111) (3211) (4211)
%e A371178 (111111) (22111) (5111)
%e A371178 (31111) (22211)
%e A371178 (211111) (32111)
%e A371178 (1111111) (221111)
%e A371178 (311111)
%e A371178 (2111111)
%e A371178 (11111111)
%t A371178 Table[Length[Select[IntegerPartitions[n],SubsetQ[#,Union@@Divisors/@#]&]],{n,0,30}]
%Y A371178 The LHS is represented by A001221, distinct case of A001222.
%Y A371178 For partitions with no divisors of parts we have A305148, ranks A316476.
%Y A371178 The RHS is represented by A370820, for prime factors A303975.
%Y A371178 The strict case is A371128.
%Y A371178 Counting all parts on the LHS gives A371130, ranks A370802.
%Y A371178 The complement is counted by A371132.
%Y A371178 For submultisets instead of distinct parts we have A371172, ranks A371165.
%Y A371178 These partitions have ranks A371177.
%Y A371178 A000005 counts divisors.
%Y A371178 A000041 counts integer partitions, strict A000009.
%Y A371178 A008284 counts partitions by length.
%Y A371178 Cf. A000837, A003963, A239312, A285573, A305148, A319055, A355529, A370803, A370808, A370813, A371168, A371171, A371173.
%K A371178 nonn
%O A371178 0,4
%A A371178 _Gus Wiseman_, Mar 17 2024