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%I #9 Jun 20 2020 01:11:52
%S 8,16,24,27,30,32,40,48,54,56,60,64,72,80,81,88,96,104,105,108,110,
%T 112,120,125,128,135,136,144,150,152,160,162,168,176,184,189,192,200,
%U 208,210,216,220,224,232,238,240,243,248,250,256,264,270,272,273,280,288
%N Numbers with three consecutive prime indices in arithmetic progression.
%C Also numbers whose first differences of prime indices do not form an anti-run, meaning there are adjacent equal differences.
%C 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.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Arithmetic_progression">Arithmetic progression</a>
%e The sequence of terms together with their prime indices begins:
%e 8: {1,1,1} 105: {2,3,4}
%e 16: {1,1,1,1} 108: {1,1,2,2,2}
%e 24: {1,1,1,2} 110: {1,3,5}
%e 27: {2,2,2} 112: {1,1,1,1,4}
%e 30: {1,2,3} 120: {1,1,1,2,3}
%e 32: {1,1,1,1,1} 125: {3,3,3}
%e 40: {1,1,1,3} 128: {1,1,1,1,1,1,1}
%e 48: {1,1,1,1,2} 135: {2,2,2,3}
%e 54: {1,2,2,2} 136: {1,1,1,7}
%e 56: {1,1,1,4} 144: {1,1,1,1,2,2}
%e 60: {1,1,2,3} 150: {1,2,3,3}
%e 64: {1,1,1,1,1,1} 152: {1,1,1,8}
%e 72: {1,1,1,2,2} 160: {1,1,1,1,1,3}
%e 80: {1,1,1,1,3} 162: {1,2,2,2,2}
%e 81: {2,2,2,2} 168: {1,1,1,2,4}
%e 88: {1,1,1,5} 176: {1,1,1,1,5}
%e 96: {1,1,1,1,1,2} 184: {1,1,1,9}
%e 104: {1,1,1,6} 189: {2,2,2,4}
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t Select[Range[100],MatchQ[Differences[primeMS[#]],{___,x_,x_,___}]&]
%Y Anti-run compositions are counted by A003242.
%Y Normal anti-runs of length n + 1 are counted by A005649.
%Y Strict partitions with equal differences are A049980.
%Y Partitions with equal differences are A049988.
%Y These are the Heinz numbers of the partitions *not* counted by A238424.
%Y Permutations avoiding triples in arithmetic progression are A295370.
%Y Strict partitions avoiding triples in arithmetic progression are A332668.
%Y Anti-run compositions are ranked by A333489.
%Y Cf. A006560, A007862, A238423, A307824, A325328, A325849, A325852.
%K nonn
%O 1,1
%A _Gus Wiseman_, Mar 29 2020
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