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Divisor-minimal numbers whose prime indices of prime indices contradict a strict version of the axiom of choice.
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%I #5 Dec 29 2023 10:56:51

%S 2,9,21,25,49,57,115,121,133,159,195,289,361,371,393,455,507,515,529,

%T 555,845,897,915,917,933,957,961,1007,1067,1183,1235,1295,1335,1443,

%U 1681,2093,2095,2135,2157,2177,2193,2197,2233,2265,2343,2369,2379,2405,2489

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

%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.

%C 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 Wikipedia, <a href="https://en.wikipedia.org/wiki/Axiom_of_choice">Axiom of choice</a>.

%e The terms together with their prime indices begin:

%e 2: {1}

%e 9: {2,2}

%e 21: {2,4}

%e 25: {3,3}

%e 49: {4,4}

%e 57: {2,8}

%e 115: {3,9}

%e 121: {5,5}

%e 133: {4,8}

%e 159: {2,16}

%e 195: {2,3,6}

%e 289: {7,7}

%e 361: {8,8}

%e 371: {4,16}

%e 393: {2,32}

%e 455: {3,4,6}

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

%t vmin[y_]:=Select[y,Function[s, Select[DeleteCases[y,s], Divisible[s,#]&]=={}]];

%t Select[Range[100],Select[Tuples[prix /@ prix[#]],UnsameQ@@#&]=={}&]//vmin

%Y The version for BII-numbers of set-systems is A368532.

%Y A000110 counts set partitions, non-isomorphic A000041.

%Y A003465 counts covering set-systems, unlabeled A055621.

%Y A007716 counts non-isomorphic multiset partitions, connected A007718.

%Y Cf. A134964, A140637, A355529, A367905, A367907.

%Y Cf. A367867, A367903, A368094, A368097, A368413, A368421.

%K nonn

%O 1,1

%A _Gus Wiseman_, Dec 29 2023