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A368187 Divisor-minimal numbers whose prime indices of prime indices contradict a strict version of the axiom of choice.

%I A368187 #5 Dec 29 2023 10:56:51
%S A368187 2,9,21,25,49,57,115,121,133,159,195,289,361,371,393,455,507,515,529,
%T A368187 555,845,897,915,917,933,957,961,1007,1067,1183,1235,1295,1335,1443,
%U A368187 1681,2093,2095,2135,2157,2177,2193,2197,2233,2265,2343,2369,2379,2405,2489
%N A368187 Divisor-minimal numbers whose prime indices of prime indices contradict a strict version of the axiom of choice.
%C A368187 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.
%C A368187 The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.
%H A368187 Wikipedia, <a href="https://en.wikipedia.org/wiki/Axiom_of_choice">Axiom of choice</a>.
%e A368187 The terms together with their prime indices begin:
%e A368187      2: {1}
%e A368187      9: {2,2}
%e A368187     21: {2,4}
%e A368187     25: {3,3}
%e A368187     49: {4,4}
%e A368187     57: {2,8}
%e A368187    115: {3,9}
%e A368187    121: {5,5}
%e A368187    133: {4,8}
%e A368187    159: {2,16}
%e A368187    195: {2,3,6}
%e A368187    289: {7,7}
%e A368187    361: {8,8}
%e A368187    371: {4,16}
%e A368187    393: {2,32}
%e A368187    455: {3,4,6}
%t A368187 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A368187 vmin[y_]:=Select[y,Function[s, Select[DeleteCases[y,s], Divisible[s,#]&]=={}]];
%t A368187 Select[Range[100],Select[Tuples[prix /@ prix[#]],UnsameQ@@#&]=={}&]//vmin
%Y A368187 The version for BII-numbers of set-systems is A368532.
%Y A368187 A000110 counts set partitions, non-isomorphic A000041.
%Y A368187 A003465 counts covering set-systems, unlabeled A055621.
%Y A368187 A007716 counts non-isomorphic multiset partitions, connected A007718.
%Y A368187 Cf. A134964, A140637, A355529, A367905, A367907.
%Y A368187 Cf. A367867, A367903, A368094, A368097, A368413, A368421.
%K A368187 nonn
%O A368187 1,1
%A A368187 _Gus Wiseman_, Dec 29 2023