cp's OEIS Frontend

A343381 Number of strict integer partitions of n with a part dividing all the others but no part divisible by all the others.

%I A343381 #6 Apr 16 2021 15:46:11
%S A343381 1,0,0,0,0,0,1,0,2,1,3,3,6,4,9,9,14,14,20,20,30,30,39,44,59,59,77,85,
%T A343381 106,114,145,150,191,205,247,267,328,345,418,455,544,582,699,745,886,
%U A343381 962,1117,1209,1430,1523,1778,1932,2225,2406,2792,3001,3456,3750
%N A343381 Number of strict integer partitions of n with a part dividing all the others but no part divisible by all the others.
%C A343381 Alternative name: Number of strict integer partitions of n that are empty or (1) have smallest part dividing all the others and (2) have greatest part not divisible by all the others.
%e A343381 The a(6) = 1 through a(16) = 14 partitions (empty column indicated by dot, A..D = 10..13):
%e A343381   321   .  431   531   541    641    642    751    761    861     862
%e A343381            521         721    731    651    5431   851    951     871
%e A343381                        4321   5321   741    6421   941    A41     961
%e A343381                                      831    7321   A31    B31     A42
%e A343381                                      921           B21    6531    B41
%e A343381                                      5421          6431   7431    D21
%e A343381                                                    6521   7521    6541
%e A343381                                                    7421   9321    7531
%e A343381                                                    8321   54321   7621
%e A343381                                                                   8431
%e A343381                                                                   8521
%e A343381                                                                   9421
%e A343381                                                                   A321
%e A343381                                                                   64321
%t A343381 Table[Length[Select[IntegerPartitions[n],#=={}||UnsameQ@@#&&And@@IntegerQ/@(#/Min@@#)&&!And@@IntegerQ/@(Max@@#/#)&]],{n,0,30}]
%Y A343381 The first condition alone gives A097986.
%Y A343381 The non-strict version is A343345 (Heinz numbers: A343340).
%Y A343381 The second condition alone gives A343377.
%Y A343381 The half-opposite versions are A343378 and A343379.
%Y A343381 The opposite (and dual) version is A343380.
%Y A343381 A000005 counts divisors.
%Y A343381 A000009 counts strict partitions.
%Y A343381 A000070 counts partitions with a selected part.
%Y A343381 A006128 counts partitions with a selected position.
%Y A343381 A015723 counts strict partitions with a selected part.
%Y A343381 A018818 counts partitions into divisors (strict: A033630).
%Y A343381 A167865 counts strict chains of divisors > 1 summing to n.
%Y A343381 A339564 counts factorizations with a selected factor.
%Y A343381 Cf. A083710, A130689, A200745, A264401, A339563, A341450, A343337, A343341, A343347, A343382.
%K A343381 nonn
%O A343381 0,9
%A A343381 _Gus Wiseman_, Apr 16 2021