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Also numbers whose first differences of prime indices do not form an anti-run, meaning there are adjacent equal differences.

Anti-run compositions are counted by A003242.

Normal anti-runs of length n + 1 are counted by A005649.

Anti-run compositions are ranked by A333489.

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These are the Heinz numbers of the partitions *not * counted by A238424.

Permutations without three consecutive parts avoiding triples in arithmetic progression are A295370.

Strict partitions without three consecutive parts avoiding triples in arithmetic progression are A332668.

allocated for Gus WisemanNumbers with three consecutive prime indices in arithmetic progression.

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

1,1

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.

Wikipedia, <a href="https://en.wikipedia.org/wiki/Arithmetic_progression">Arithmetic progression</a>

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}

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

Select[Range[100], MatchQ[Differences[primeMS[#]], {___, x_, x_, ___}]&]

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 without three consecutive parts in arithmetic progression are A295370.

Strict partitions without three consecutive parts in arithmetic progression are A332668.

Cf. A006560, A007862, A238423, A307824, A325328, A325849, A325852.

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Gus Wiseman, Mar 29 2020

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allocated for Gus Wiseman

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