A370348 Numbers k such that there are fewer divisors of prime indices of k than there are prime indices of k.

4, 8, 12, 16, 18, 20, 24, 27, 32, 36, 40, 44, 48, 50, 54, 56, 60, 64, 68, 72, 80, 81, 84, 88, 90, 96, 100, 108, 112, 120, 124, 125, 126, 128, 132, 135, 136, 144, 150, 160, 162, 164, 168, 176, 180, 184, 189, 192, 196, 198, 200, 204, 208, 216, 220, 224, 225, 236, 240, 242, 243, 248, 250, 252, 256

Offset

1,1

Comments

No multiple of a term is a term of A368110.

Links

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

Example

a(5) = 18 is a term because the prime indices of 18 = 2 * 3^2 are 1,2,2, and there are 3 of these but only 2 divisors of prime indices, namely 1 and 2.

Maple

filter:= proc(n) uses numtheory; local F, D, t;
   F:= map(t -> [pi(t[1]), t[2]], ifactors(n)[2]);
   D:= `union`(seq(divisors(t[1]), t = F));
   nops(D) < add(t[2], t = F)
end proc:
select(filter, [$1..300]);

Mathematica

filter[n_] := Module[{F, d},
    F = {PrimePi[#[[1]]], #[[2]]}& /@ FactorInteger[n];
    d = Union[Flatten[Divisors /@ F[[All, 1]]]];
    Length[d] < Total[F[[All, 2]]]];
Select[Range[300], filter] (* Jean-François Alcover, Mar 08 2024, after Maple code *)

Crossrefs

The LHS is A370820, firsts A371131.

The version for equality is A370802, counted by A371130, strict A371128.

For submultisets instead of parts on the RHS we get A371167.

The opposite version is A371168, counted by A371173.

The weak version is A371169.

The complement is A371170.

Partitions of this type are counted by A371171.

A000005 counts divisors.

A001221 counts distinct prime factors.

A027746 lists prime factors, indices A112798, length A001222.

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

Cf. A003963, A239312, A303975, A319899, A355529, A355739, A370808, A371127.

Sequence in context: A130702 A355740 A371089 * A053806 A068306 A311118

Adjacent sequences: A370345 A370346 A370347 * A370349 A370350 A370351

Keyword

nonn

Author

Robert Israel, Feb 15 2024

Status

approved