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⇱ A389812 - OEIS

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A389812
Numbers whose prime indices are not in arithmetic progression.
10
12, 18, 20, 24, 28, 36, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 63, 66, 68, 70, 72, 75, 76, 78, 80, 84, 88, 90, 92, 96, 98, 99, 100, 102, 104, 108, 112, 114, 116, 117, 120, 124, 126, 130, 132, 135, 136, 138, 140, 144, 147, 148, 150, 152, 153, 154, 156, 160
OFFSET
1,1
COMMENTS
These are numbers whose first differences of prime indices are not all equal, counted by A389811. 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.
EXAMPLE
The terms together with their prime indices begin:
12: {1,1,2}
18: {1,2,2}
20: {1,1,3}
24: {1,1,1,2}
28: {1,1,4}
36: {1,1,2,2}
40: {1,1,1,3}
42: {1,2,4}
44: {1,1,5}
45: {2,2,3}
48: {1,1,1,1,2}
50: {1,3,3}
52: {1,1,6}
54: {1,2,2,2}
56: {1,1,1,4}
60: {1,1,2,3}
63: {2,2,4}
66: {1,2,5}
68: {1,1,7}
70: {1,3,4}
72: {1,1,1,2,2}
75: {2,3,3}
MATHEMATICA
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], !SameQ@@Differences[prix[#]]&]
CROSSREFS
The 0-appended complement is A325327.
The complement is A325328, counted by A049988.
The complement for distinct differences is A325368, counted by A325325.
The complement for compositions is A389731, counted by A175342.
For compositions instead of partitions we have A389741, ranks A389735.
These partitions are counted by A389811.
For distinct instead of equal differences we have A389919, ranks A389599.
A000041 counts partitions.
A000837 counts aperiodic partitions.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A389597 ranks compositions with distinct differences, counted by A325545.
Sequence in context: A036456 A102467 A387723 * A342973 A126706 A123711
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 21 2025
STATUS
approved