A355740 Numbers of which it is not possible to choose a different divisor of each prime index.

4, 8, 12, 16, 18, 20, 24, 27, 28, 32, 36, 40, 44, 48, 50, 52, 54, 56, 60, 64, 68, 72, 76, 80, 81, 84, 88, 90, 92, 96, 100, 104, 108, 112, 116, 120, 124, 125, 126, 128, 132, 135, 136, 140, 144, 148, 150, 152, 156, 160, 162, 164, 168, 172, 176, 180, 184, 188

Offset

1,1

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 a sub-multiset S of the prime indices of k such that fewer than |S| numbers are divisors of a member of S. Equivalently, k is divisible by a member of A370348. - Robert Israel, Feb 15 2024

Links

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

Wikipedia, Cartesian product.

Formula

We have A001221(a(n)) >= A303975(a(n)).

Example

The terms together with their prime indices begin:
    4: {1,1}
    8: {1,1,1}
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   24: {1,1,1,2}
   27: {2,2,2}
   28: {1,1,4}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   40: {1,1,1,3}
   44: {1,1,5}
   48: {1,1,1,1,2}
For example, the choices of a divisor of each prime index of 90 are: (1,1,1,1), (1,1,1,3), (1,1,2,1), (1,1,2,3), (1,2,1,1), (1,2,1,3), (1,2,2,1), (1,2,2,3). But none of these has all distinct elements, so 90 is in the sequence.

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:
remove(filter, [$1..200]); # Robert Israel, Feb 15 2024

Mathematica

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

Crossrefs

Positions of 0's in A355739.

The case of just prime factors (not all divisors) is A355529, odd A355535.

The unordered case is counted by A355733, firsts A355734.

A000005 counts divisors.

A001414 adds up distinct prime divisors, counted by A001221.

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, A335433, A335448, A340827, A355737, A355749, A370348.

Sequence in context: A274919 A196032 A130702 * A371089 A370348 A053806

Adjacent sequences: A355737 A355738 A355739 * A355741 A355742 A355743

Keyword

nonn

Author

Gus Wiseman, Jul 22 2022

Status

approved