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

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, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97

Offset: 1

Author

Gus Wiseman, Dec 15 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.

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

Examples

			The terms together with their prime indices begin:
   1: {}
   2: {1}
   3: {2}
   5: {3}
   6: {1,2}
   7: {4}
   9: {2,2}
  10: {1,3}
  11: {5}
  13: {6}
  14: {1,4}
  15: {2,3}
  17: {7}
  19: {8}
  21: {2,4}
  22: {1,5}
  23: {9}
  25: {3,3}
  26: {1,6}
  29: {10}
  30: {1,2,3}
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Partitions of this type are counted by A239312, complement A370320.

Positions of nonzero terms in A355739.

Complement of A355740.

For just prime divisors we have A368100, complement A355529 (odd A355535).

A000005 counts divisors.

A003963 multiplies together the prime indices of n.

A056239 adds up prime indices, row sums of A112798, counted by A001222.

A120383 lists numbers divisible by all of their prime indices.

A324850 lists numbers divisible by the product of their prime indices.

A355731 counts choices of a divisor of each prime index, firsts A355732.

A355741 chooses prime factors of prime indices, variations A355744, A355745.

Cf. A000720, A076610, A111774, A335433, A335448, A340852, A355733, A355734, A355737, A355749, A370348.

Programs

Maple

filter:= proc(n) uses numtheory, GraphTheory; local B,S,F,D,E,G,t,d;
  F:= ifactors(n)[2];
  F:= map(t -> [pi(t[1]),t[2]], F);
  D:= `union`(seq(divisors(t[1]), t = F));
  F:= map(proc(t) local i;seq([t[1],i],i=1..t[2]) end proc,F);
  if nops(D) < nops(F) then return false fi;
  E:= {seq(seq({t,d},d=divisors(t[1])),t = F)};
  S:= map(t -> convert(t,name), [op(F),op(D)]);
  E:= map(e -> map(convert,e,name),E);
  G:= Graph(S,E);
  B:= BipartiteMatching(G);
  B[1] = nops(F);
end proc:
select(filter, [$1..100]); # Robert Israel, Feb 15 2024

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Select[Tuples[Divisors/@prix[#]],UnsameQ@@#&]!={}&]

Formula

Heinz numbers of the partitions counted by A239312.