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A333195
Numbers with three consecutive prime indices in arithmetic progression.
3
8, 16, 24, 27, 30, 32, 40, 48, 54, 56, 60, 64, 72, 80, 81, 88, 96, 104, 105, 108, 110, 112, 120, 125, 128, 135, 136, 144, 150, 152, 160, 162, 168, 176, 184, 189, 192, 200, 208, 210, 216, 220, 224, 232, 238, 240, 243, 248, 250, 256, 264, 270, 272, 273, 280, 288
OFFSET
1,1
COMMENTS
Also numbers whose first differences of prime indices do not form an anti-run, meaning there are adjacent equal differences.
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 sequence of terms together with their prime indices begins:
8: {1,1,1} 105: {2,3,4}
16: {1,1,1,1} 108: {1,1,2,2,2}
24: {1,1,1,2} 110: {1,3,5}
27: {2,2,2} 112: {1,1,1,1,4}
30: {1,2,3} 120: {1,1,1,2,3}
32: {1,1,1,1,1} 125: {3,3,3}
40: {1,1,1,3} 128: {1,1,1,1,1,1,1}
48: {1,1,1,1,2} 135: {2,2,2,3}
54: {1,2,2,2} 136: {1,1,1,7}
56: {1,1,1,4} 144: {1,1,1,1,2,2}
60: {1,1,2,3} 150: {1,2,3,3}
64: {1,1,1,1,1,1} 152: {1,1,1,8}
72: {1,1,1,2,2} 160: {1,1,1,1,1,3}
80: {1,1,1,1,3} 162: {1,2,2,2,2}
81: {2,2,2,2} 168: {1,1,1,2,4}
88: {1,1,1,5} 176: {1,1,1,1,5}
96: {1,1,1,1,1,2} 184: {1,1,1,9}
104: {1,1,1,6} 189: {2,2,2,4}
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], MatchQ[Differences[primeMS[#]], {___, x_, x_, ___}]&]
CROSSREFS
Anti-run compositions are counted by A003242.
Normal anti-runs of length n + 1 are counted by A005649.
Strict partitions with equal differences are A049980.
Partitions with equal differences are A049988.
These are the Heinz numbers of the partitions *not* counted by A238424.
Permutations avoiding triples in arithmetic progression are A295370.
Strict partitions avoiding triples in arithmetic progression are A332668.
Anti-run compositions are ranked by A333489.
Sequence in context: A282256 A037370 A325261 * A229972 A144591 A078131
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 29 2020
STATUS
approved