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

2, 9, 21, 25, 49, 57, 115, 121, 133, 159, 195, 289, 361, 371, 393, 455, 507, 515, 529, 555, 845, 897, 915, 917, 933, 957, 961, 1007, 1067, 1183, 1235, 1295, 1335, 1443, 1681, 2093, 2095, 2135, 2157, 2177, 2193, 2197, 2233, 2265, 2343, 2369, 2379, 2405, 2489

Offset: 1

Author

Gus Wiseman, Dec 29 2023

Keywords

• nonn

Comments

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 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.

Examples

			The terms together with their prime indices begin:
     2: {1}
     9: {2,2}
    21: {2,4}
    25: {3,3}
    49: {4,4}
    57: {2,8}
   115: {3,9}
   121: {5,5}
   133: {4,8}
   159: {2,16}
   195: {2,3,6}
   289: {7,7}
   361: {8,8}
   371: {4,16}
   393: {2,32}
   455: {3,4,6}
		

Links

• Wikipedia, Axiom of choice.

Crossrefs

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

A000110 counts set partitions, non-isomorphic A000041.

A003465 counts covering set-systems, unlabeled A055621.

A007716 counts non-isomorphic multiset partitions, connected A007718.

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

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

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
vmin[y_]:=Select[y,Function[s, Select[DeleteCases[y,s], Divisible[s,#]&]=={}]];
Select[Range[100],Select[Tuples[prix /@ prix[#]],UnsameQ@@#&]=={}&]//vmin