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A333195 Numbers with three consecutive prime indices in arithmetic progression.

%I A333195 #9 Jun 20 2020 01:11:52
%S A333195 8,16,24,27,30,32,40,48,54,56,60,64,72,80,81,88,96,104,105,108,110,
%T A333195 112,120,125,128,135,136,144,150,152,160,162,168,176,184,189,192,200,
%U A333195 208,210,216,220,224,232,238,240,243,248,250,256,264,270,272,273,280,288
%N A333195 Numbers with three consecutive prime indices in arithmetic progression.
%C A333195 Also numbers whose first differences of prime indices do not form an anti-run, meaning there are adjacent equal differences.
%C A333195 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 A333195 Wikipedia, <a href="https://en.wikipedia.org/wiki/Arithmetic_progression">Arithmetic progression</a>
%e A333195 The sequence of terms together with their prime indices begins:
%e A333195     8: {1,1,1}          105: {2,3,4}
%e A333195    16: {1,1,1,1}        108: {1,1,2,2,2}
%e A333195    24: {1,1,1,2}        110: {1,3,5}
%e A333195    27: {2,2,2}          112: {1,1,1,1,4}
%e A333195    30: {1,2,3}          120: {1,1,1,2,3}
%e A333195    32: {1,1,1,1,1}      125: {3,3,3}
%e A333195    40: {1,1,1,3}        128: {1,1,1,1,1,1,1}
%e A333195    48: {1,1,1,1,2}      135: {2,2,2,3}
%e A333195    54: {1,2,2,2}        136: {1,1,1,7}
%e A333195    56: {1,1,1,4}        144: {1,1,1,1,2,2}
%e A333195    60: {1,1,2,3}        150: {1,2,3,3}
%e A333195    64: {1,1,1,1,1,1}    152: {1,1,1,8}
%e A333195    72: {1,1,1,2,2}      160: {1,1,1,1,1,3}
%e A333195    80: {1,1,1,1,3}      162: {1,2,2,2,2}
%e A333195    81: {2,2,2,2}        168: {1,1,1,2,4}
%e A333195    88: {1,1,1,5}        176: {1,1,1,1,5}
%e A333195    96: {1,1,1,1,1,2}    184: {1,1,1,9}
%e A333195   104: {1,1,1,6}        189: {2,2,2,4}
%t A333195 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A333195 Select[Range[100],MatchQ[Differences[primeMS[#]],{___,x_,x_,___}]&]
%Y A333195 Anti-run compositions are counted by A003242.
%Y A333195 Normal anti-runs of length n + 1 are counted by A005649.
%Y A333195 Strict partitions with equal differences are A049980.
%Y A333195 Partitions with equal differences are A049988.
%Y A333195 These are the Heinz numbers of the partitions *not* counted by A238424.
%Y A333195 Permutations avoiding triples in arithmetic progression are A295370.
%Y A333195 Strict partitions avoiding triples in arithmetic progression are A332668.
%Y A333195 Anti-run compositions are ranked by A333489.
%Y A333195 Cf. A006560, A007862, A238423, A307824, A325328, A325849, A325852.
%K A333195 nonn
%O A333195 1,1
%A A333195 _Gus Wiseman_, Mar 29 2020