cp's OEIS Frontend

A341450 Number of strict integer partitions of n that are empty or have smallest part not dividing all the others.

%I A341450 #9 Apr 15 2021 08:18:31
%S A341450 1,0,0,0,0,1,0,2,1,3,3,6,3,9,9,12,12,20,18,28,27,37,42,55,51,74,80,98,
%T A341450 105,136,137,180,189,232,255,308,320,403,434,512,551,668,706,852,915,
%U A341450 1067,1170,1370,1453,1722,1860,2145,2332,2701,2899,3355,3626,4144
%N A341450 Number of strict integer partitions of n that are empty or have smallest part not dividing all the others.
%C A341450 Alternative name: Number of strict integer partitions of n with no part dividing all the others.
%F A341450 a(n > 0) = A000009(n) - Sum_{d|n} A025147(d-1).
%e A341450 The a(0) = 1 through a(15) = 12 strict partitions (empty columns indicated by dots, 0 represents the empty partition, A..D = 10..13):
%e A341450   0  .  .  .  .  32   .  43   53   54    64    65    75    76    86     87
%e A341450                          52        72    73    74    543   85    95     96
%e A341450                                    432   532   83    732   94    A4     B4
%e A341450                                                92          A3    B3     D2
%e A341450                                                542         B2    653    654
%e A341450                                                632         643   743    753
%e A341450                                                            652   752    762
%e A341450                                                            742   932    843
%e A341450                                                            832   5432   852
%e A341450                                                                         942
%e A341450                                                                         A32
%e A341450                                                                         6432
%t A341450 Table[Length[Select[IntegerPartitions[n],#=={}||UnsameQ@@#&&!And@@IntegerQ/@(#/Min@@#)&]],{n,0,30}]
%Y A341450 The complement is counted by A097986 (non-strict: A083710, rank: A339563).
%Y A341450 The complement with no 1's is A098965 (non-strict: A083711).
%Y A341450 The non-strict version is A338470.
%Y A341450 The Heinz numbers of these partitions are A339562 (non-strict: A342193).
%Y A341450 The case with greatest part not divisible by all others is A343379.
%Y A341450 The case with greatest part divisible by all others is A343380.
%Y A341450 A000009 counts strict partitions (non-strict: A000041).
%Y A341450 A000070 counts partitions with a selected part.
%Y A341450 A006128 counts partitions with a selected position.
%Y A341450 A015723 counts strict partitions with a selected part.
%Y A341450 A167865 counts strict chains of divisors > 1 summing to n.
%Y A341450 Sequences with similar formulas: A024994, A047966, A047968, A168111.
%Y A341450 Cf. A001787, A001792, A064410, A264401, A343342, A343381, A343382.
%K A341450 nonn
%O A341450 0,8
%A A341450 _Gus Wiseman_, Apr 15 2021