cp's OEIS Frontend

A368110 Numbers of which it is possible to choose a different divisor of each prime index.

%I A368110 #18 Mar 23 2024 22:12:40
%S A368110 1,2,3,5,6,7,9,10,11,13,14,15,17,19,21,22,23,25,26,29,30,31,33,34,35,
%T A368110 37,38,39,41,42,43,45,46,47,49,51,53,55,57,58,59,61,62,63,65,66,67,69,
%U A368110 70,71,73,74,75,77,78,79,82,83,85,86,87,89,91,93,94,95,97
%N A368110 Numbers of which it is possible to choose a different divisor of each prime index.
%C A368110 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 A368110 By Hall's marriage theorem, k is a term if and only if there is no sub-multiset S of the prime indices of k such that fewer than |S| numbers are divisors of a member of S. Equivalently, there is no divisor of k in A370348. - _Robert Israel_, Feb 15 2024
%H A368110 Robert Israel, <a href="/A368110/b368110.txt">Table of n, a(n) for n = 1..10000</a>
%F A368110 Heinz numbers of the partitions counted by A239312.
%e A368110 The terms together with their prime indices begin:
%e A368110    1: {}
%e A368110    2: {1}
%e A368110    3: {2}
%e A368110    5: {3}
%e A368110    6: {1,2}
%e A368110    7: {4}
%e A368110    9: {2,2}
%e A368110   10: {1,3}
%e A368110   11: {5}
%e A368110   13: {6}
%e A368110   14: {1,4}
%e A368110   15: {2,3}
%e A368110   17: {7}
%e A368110   19: {8}
%e A368110   21: {2,4}
%e A368110   22: {1,5}
%e A368110   23: {9}
%e A368110   25: {3,3}
%e A368110   26: {1,6}
%e A368110   29: {10}
%e A368110   30: {1,2,3}
%p A368110 filter:= proc(n) uses numtheory, GraphTheory; local B,S,F,D,E,G,t,d;
%p A368110   F:= ifactors(n)[2];
%p A368110   F:= map(t -> [pi(t[1]),t[2]], F);
%p A368110   D:= `union`(seq(divisors(t[1]), t = F));
%p A368110   F:= map(proc(t) local i;seq([t[1],i],i=1..t[2]) end proc,F);
%p A368110   if nops(D) < nops(F) then return false fi;
%p A368110   E:= {seq(seq({t,d},d=divisors(t[1])),t = F)};
%p A368110   S:= map(t -> convert(t,name), [op(F),op(D)]);
%p A368110   E:= map(e -> map(convert,e,name),E);
%p A368110   G:= Graph(S,E);
%p A368110   B:= BipartiteMatching(G);
%p A368110   B[1] = nops(F);
%p A368110 end proc:
%p A368110 select(filter, [$1..100]); # _Robert Israel_, Feb 15 2024
%t A368110 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A368110 Select[Range[100],Select[Tuples[Divisors/@prix[#]],UnsameQ@@#&]!={}&]
%Y A368110 Partitions of this type are counted by A239312, complement A370320.
%Y A368110 Positions of nonzero terms in A355739.
%Y A368110 Complement of A355740.
%Y A368110 For just prime divisors we have A368100, complement A355529 (odd A355535).
%Y A368110 A000005 counts divisors.
%Y A368110 A003963 multiplies together the prime indices of n.
%Y A368110 A056239 adds up prime indices, row sums of A112798, counted by A001222.
%Y A368110 A120383 lists numbers divisible by all of their prime indices.
%Y A368110 A324850 lists numbers divisible by the product of their prime indices.
%Y A368110 A355731 counts choices of a divisor of each prime index, firsts A355732.
%Y A368110 A355741 chooses prime factors of prime indices, variations A355744, A355745.
%Y A368110 Cf. A000720, A076610, A111774, A335433, A335448, A340852, A355733, A355734, A355737, A355749, A370348.
%K A368110 nonn
%O A368110 1,2
%A A368110 _Gus Wiseman_, Dec 15 2023