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A341450 Number of strict integer partitions of n that are empty or have smallest part not dividing all the others. 25
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, 105, 136, 137, 180, 189, 232, 255, 308, 320, 403, 434, 512, 551, 668, 706, 852, 915, 1067, 1170, 1370, 1453, 1722, 1860, 2145, 2332, 2701, 2899, 3355, 3626, 4144 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,8
COMMENTS
Alternative name: Number of strict integer partitions of n with no part dividing all the others.
LINKS
FORMULA
a(n > 0) = A000009(n) - Sum_{d|n} A025147(d-1).
EXAMPLE
The a(0) = 1 through a(15) = 12 strict partitions (empty columns indicated by dots, 0 represents the empty partition, A..D = 10..13):
0 . . . . 32 . 43 53 54 64 65 75 76 86 87
52 72 73 74 543 85 95 96
432 532 83 732 94 A4 B4
92 A3 B3 D2
542 B2 653 654
632 643 743 753
652 752 762
742 932 843
832 5432 852
942
A32
6432
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], #=={}||UnsameQ@@#&&!And@@IntegerQ/@(#/Min@@#)&]], {n, 0, 30}]
CROSSREFS
The complement is counted by A097986 (non-strict: A083710, rank: A339563).
The complement with no 1's is A098965 (non-strict: A083711).
The non-strict version is A338470.
The Heinz numbers of these partitions are A339562 (non-strict: A342193).
The case with greatest part not divisible by all others is A343379.
The case with greatest part divisible by all others is A343380.
A000009 counts strict partitions (non-strict: A000041).
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A167865 counts strict chains of divisors > 1 summing to n.
Sequences with similar formulas: A024994, A047966, A047968, A168111.
Sequence in context: A213934 A124774 A056610 * A343381 A336096 A227774
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 15 2021
STATUS
approved

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Last modified March 19 01:57 EDT 2024. Contains 370952 sequences. (Running on oeis4.)