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 A343344 #6 Apr 15 2021 21:43:02 %S A343344 1,0,0,0,0,0,0,0,0,0,0,1,0,1,1,1,1,5,1,6,4,6,7,15,6,16,15,20,17,36,18, %T A343344 43,36,46,48,72,45,93,82,103,88,152,104,179,158,191,194,285,202,328, %U A343344 292,373,348,502,391,576,519,659,634,864,665 %N A343344 Number of integer partitions of n that are either empty, or do not have smallest part dividing all the others, but do have greatest part divisible by all the others. %C A343344 Alternative name: Number of integer partitions of n with no part dividing all the others, but with a part divisible by all the others. %e A343344 The a(18) = 1 through a(23) = 15 partitions (A..E = 10..14): %e A343344 633222 C43 C332 C432 C64 E72 %e A343344 A522 66332 A5222 A552 F53 %e A343344 C322 633332 C3222 C433 I32 %e A343344 66322 6332222 663222 C3322 C443 %e A343344 633322 6333222 663322 C632 %e A343344 6322222 63222222 6333322 66632 %e A343344 63322222 C3332 %e A343344 C4322 %e A343344 663332 %e A343344 A52222 %e A343344 C32222 %e A343344 6333332 %e A343344 6632222 %e A343344 63332222 %e A343344 632222222 %t A343344 Table[Length[Select[IntegerPartitions[n],#=={}||!And@@IntegerQ/@(#/Min@@#)&&And@@IntegerQ/@(Max@@#/#)&]],{n,0,30}] %Y A343344 The second condition alone gives A130689. %Y A343344 The half-opposite versions are A130714 and A343342. %Y A343344 The first condition alone gives A338470. %Y A343344 The Heinz numbers of these partitions are 1 and A343339. %Y A343344 The opposite version is A343345. %Y A343344 The strict case is A343380. %Y A343344 A000009 counts strict partitions. %Y A343344 A000041 counts partitions. %Y A343344 A000070 counts partitions with a selected part. %Y A343344 A006128 counts partitions with a selected position. %Y A343344 A015723 counts strict partitions with a selected part. %Y A343344 Cf. A083710, A097986, A264401, A339562, A341450, A342193, A343346. %K A343344 nonn %O A343344 0,18 %A A343344 _Gus Wiseman_, Apr 15 2021