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%I #5 Feb 19 2019 00:08:09
%S 2117,3973,4234,4843,5183,5249,5891,6351,6757,7181,7801,7946,8249,
%T 8468,8903,9193,9686,9727,10019,10063,10366,10498,10585,11051,11513,
%U 11567,11782,11857,11919,12557,12629,12702,12851,13021,13193,13459,13514,13631,14123,14362
%N Numbers whose multiset multisystem (A302242) is crossing.
%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. The multiset multisystem of n is obtained by taking the multiset of prime indices of each prime index of n.
%C A multiset of multisets is crossing if it contains a 2-element submultiset of the form {{...x...y...}, {...z...t...}} where x < z < y < t or z < x < t < y.
%e The sequence of terms together with their multiset multisystems begins:
%e 2117: {{1,3},{2,4}}
%e 3973: {{1,3},{2,5}}
%e 4234: {{},{1,3},{2,4}}
%e 4843: {{1,3},{2,6}}
%e 5183: {{1,1,3},{2,4}}
%e 5249: {{1,3},{1,2,4}}
%e 5891: {{1,4},{2,5}}
%e 6351: {{1},{1,3},{2,4}}
%e 6757: {{1,3},{2,7}}
%e 7181: {{1,4},{2,6}}
%e 7801: {{1,3},{2,8}}
%e 7946: {{},{1,3},{2,5}}
%e 8249: {{2,4},{1,2,3}}
%e 8468: {{},{},{1,3},{2,4}}
%e 8903: {{1,3},{2,2,4}}
%e 9193: {{1,3},{1,2,5}}
%e 9686: {{},{1,3},{2,6}}
%e 9727: {{1,1,3},{2,5}}
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t croXQ[stn_]:=MatchQ[stn,{___,{___,x_,___,y_,___},___,{___,z_,___,t_,___},___}/;x<z<y<t||z<x<t<y];
%t Select[Range[10000],croXQ[primeMS/@primeMS[#]]&]
%Y Cf. A000108, A001055, A001263, A002662, A016098, A054726, A056239, A099947, A112798, A302242.
%Y Cf. A324166, A324167, A324168, A324169, A324171, A324173.
%K nonn
%O 1,1
%A _Gus Wiseman_, Feb 17 2019
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