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A387577
Numbers k such that it is not possible to choose a sequence of distinct integer partitions, one of each prime index of k (with multiplicity).
14
4, 8, 12, 16, 20, 24, 27, 28, 32, 36, 40, 44, 48, 52, 54, 56, 60, 64, 68, 72, 76, 80, 81, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 135, 136, 140, 144, 148, 152, 156, 160, 162, 164, 168, 172, 176, 180, 184, 188, 189, 192, 196, 200, 204, 208
OFFSET
1,1
COMMENTS
First differs from A100716 in having 625.
First differs from A276079 in lacking 16807.
First differs from A328251 in having 968 and lacking 225.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
Also numbers having a prime factor prime(x) whose exponent exceeds A000041(x).
EXAMPLE
The partition (3,3,3) is not counted, as we have choices such as ((3),(1,1,1),(2,1)). However, the partition (3,3,3,3) is counted, so 625 is in the sequence but 125 is not.
The terms together with their prime indices begin:
4: {1,1}
8: {1,1,1}
12: {1,1,2}
16: {1,1,1,1}
20: {1,1,3}
24: {1,1,1,2}
27: {2,2,2}
28: {1,1,4}
32: {1,1,1,1,1}
36: {1,1,2,2}
40: {1,1,1,3}
44: {1,1,5}
48: {1,1,1,1,2}
52: {1,1,6}
54: {1,2,2,2}
56: {1,1,1,4}
60: {1,1,2,3}
64: {1,1,1,1,1,1}
MATHEMATICA
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], Select[Tuples[IntegerPartitions/@prix[#]], UnsameQ@@#&]=={}&]
CROSSREFS
For initial intervals instead of partitions we have A276079, complement A276078.
For divisors instead of partitions we have A355740, counted by A370320 (strict A387137).
The complement for divisors is A368110, counted by A239312.
These are the positions of zero in A387115.
Integer partitions of this type are counted by A387134.
For sets of strict partitions we have A387176, complement A387177.
For sets of constant partitions we have A387180, counted by A387329.
The complement is A387576, counted by A387328.
A000041 counts integer partitions, strict A000009.
A003963 multiplies together the prime indices of n.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A238873 counts superdiagonal partitions, ranks A387112.
A289509 lists numbers with relatively prime prime indices.
A387118 counts non-superdiagonal partitions, ranks A387113.
Sequence in context: A274141 A086133 A387180 * A100716 A328251 A276079
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 09 2025
STATUS
approved