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Number of strict integer partitions of n that are empty or have smallest part not dividing all the others.
25

%I #9 Apr 15 2021 08:18:31

%S 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 105,136,137,180,189,232,255,308,320,403,434,512,551,668,706,852,915,

%U 1067,1170,1370,1453,1722,1860,2145,2332,2701,2899,3355,3626,4144

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

%C Alternative name: Number of strict integer partitions of n with no part dividing all the others.

%F a(n > 0) = A000009(n) - Sum_{d|n} A025147(d-1).

%e 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 0 . . . . 32 . 43 53 54 64 65 75 76 86 87

%e 52 72 73 74 543 85 95 96

%e 432 532 83 732 94 A4 B4

%e 92 A3 B3 D2

%e 542 B2 653 654

%e 632 643 743 753

%e 652 752 762

%e 742 932 843

%e 832 5432 852

%e 942

%e A32

%e 6432

%t Table[Length[Select[IntegerPartitions[n],#=={}||UnsameQ@@#&&!And@@IntegerQ/@(#/Min@@#)&]],{n,0,30}]

%Y The complement is counted by A097986 (non-strict: A083710, rank: A339563).

%Y The complement with no 1's is A098965 (non-strict: A083711).

%Y The non-strict version is A338470.

%Y The Heinz numbers of these partitions are A339562 (non-strict: A342193).

%Y The case with greatest part not divisible by all others is A343379.

%Y The case with greatest part divisible by all others is A343380.

%Y A000009 counts strict partitions (non-strict: A000041).

%Y A000070 counts partitions with a selected part.

%Y A006128 counts partitions with a selected position.

%Y A015723 counts strict partitions with a selected part.

%Y A167865 counts strict chains of divisors > 1 summing to n.

%Y Sequences with similar formulas: A024994, A047966, A047968, A168111.

%Y Cf. A001787, A001792, A064410, A264401, A343342, A343381, A343382.

%K nonn

%O 0,8

%A _Gus Wiseman_, Apr 15 2021