cp's OEIS Frontend

A343379 Number of strict integer partitions of n with no part dividing or divisible by all the other parts.

%I A343379 #6 Apr 16 2021 15:45:57
%S A343379 1,0,0,0,0,1,0,2,1,3,3,5,3,9,9,12,12,18,18,27,27,36,41,51,51,73,80,96,
%T A343379 105,132,137,177,188,230,253,303,320,398,431,508,550,659,705,847,913,
%U A343379 1063,1165,1359,1452,1716,1856,2134,2329,2688,2894,3345,3622,4133
%N A343379 Number of strict integer partitions of n with no part dividing or divisible by all the other parts.
%C A343379 Alternative name: Number of strict integer partitions of n that are either empty, or (1) have smallest part not dividing all the others and (2) have greatest part not divisible by all the others.
%F A343379 The Heinz numbers for the non-strict version are A343338 = A342193 /\ A343337.
%e A343379 The a(5) = 1 through a(13) = 9 partitions (empty column indicated by dot):
%e A343379   (3,2)  .  (4,3)  (5,3)  (5,4)    (6,4)    (6,5)    (7,5)    (7,6)
%e A343379             (5,2)         (7,2)    (7,3)    (7,4)    (5,4,3)  (8,5)
%e A343379                           (4,3,2)  (5,3,2)  (8,3)    (7,3,2)  (9,4)
%e A343379                                             (9,2)             (10,3)
%e A343379                                             (5,4,2)           (11,2)
%e A343379                                                               (6,4,3)
%e A343379                                                               (6,5,2)
%e A343379                                                               (7,4,2)
%e A343379                                                               (8,3,2)
%t A343379 Table[Length[Select[IntegerPartitions[n],#=={}||UnsameQ@@#&&!And@@IntegerQ/@(#/Min@@#)&&!And@@IntegerQ/@(Max@@#/#)&]],{n,0,30}]
%Y A343379 The first condition alone gives A341450.
%Y A343379 The non-strict version is A343342  (Heinz numbers: A343338).
%Y A343379 The second condition alone gives A343377.
%Y A343379 The opposite version is A343378.
%Y A343379 The half-opposite versions are A343380 and A343381.
%Y A343379 The version for "or" instead of "and" is A343382.
%Y A343379 A000009 counts strict partitions.
%Y A343379 A000070 counts partitions with a selected part.
%Y A343379 A006128 counts partitions with a selected position.
%Y A343379 A015723 counts strict partitions with a selected part.
%Y A343379 A018818 counts partitions into divisors (strict: A033630).
%Y A343379 A167865 counts strict chains of divisors > 1 summing to n.
%Y A343379 A339564 counts factorizations with a selected factor.
%Y A343379 Cf. A083710, A097986, A200745, A264401, A338470, A339562, A342193, A343337, A343341, A343343, A343346, A343347.
%K A343379 nonn
%O A343379 0,8
%A A343379 _Gus Wiseman_, Apr 16 2021