A119313 Numbers with a prime as third-smallest divisor.

6, 10, 12, 14, 15, 18, 21, 22, 24, 26, 30, 33, 34, 35, 36, 38, 39, 42, 45, 46, 48, 50, 51, 54, 55, 57, 58, 60, 62, 63, 65, 66, 69, 70, 72, 74, 75, 77, 78, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 96, 98, 102, 105, 106, 108, 110, 111, 114, 115, 118, 119, 120, 122, 123, 126

Offset

1,1

Comments

m is a term iff A001221(m) > 1 and (A067029(m) = 1 or A119288(m) < A020639(m)^2).

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Example

a(1) = A087134(3) = 6.
From Gus Wiseman, Oct 19 2019: (Start)
The sequence of terms together with their divisors begins:
    6: {1,2,3,6}
   10: {1,2,5,10}
   12: {1,2,3,4,6,12}
   14: {1,2,7,14}
   15: {1,3,5,15}
   18: {1,2,3,6,9,18}
   21: {1,3,7,21}
   22: {1,2,11,22}
   24: {1,2,3,4,6,8,12,24}
   26: {1,2,13,26}
   30: {1,2,3,5,6,10,15,30}
   33: {1,3,11,33}
   34: {1,2,17,34}
   35: {1,5,7,35}
   36: {1,2,3,4,6,9,12,18,36}
   38: {1,2,19,38}
   39: {1,3,13,39}
   42: {1,2,3,6,7,14,21,42}
   45: {1,3,5,9,15,45}
   46: {1,2,23,46}
(End)

Maple

q:= n-> (l-> nops(l)>2 and isprime(l[3]))(
         sort([numtheory[divisors](n)[]])):
select(q, [$1..200])[];  # Alois P. Heinz, Oct 19 2019

Mathematica

Select[Range[100], Length[Divisors[#]]>2&&PrimeQ[Divisors[#][[3]]]&] (* Gus Wiseman, Oct 15 2019 *)
Select[Range[130], Length[f = FactorInteger[#]] > 1 && (f[[1, 2]] == 1 || f[[1, 1]]^2 > f[[2, 1]]) &] (* Amiram Eldar, Jul 02 2022 *)

Crossrefs

Complement of A119314.

Subsequences: A006881, A000469, A008588.

A subset of A002808 and A080257.

Numbers whose third-largest divisor is prime are A328338.

Second-smallest divisor is A020639.

Third-smallest divisor is A292269.

Cf. A000005, A000040, A001221, A020639, A027750, A033676, A060775, A067029, A088725, A119288, A328189.

Sequence in context: A273157 A139587 A201463 * A362980 A097318 A080363

Adjacent sequences: A119310 A119311 A119312 * A119314 A119315 A119316

Keyword

nonn

Author

Reinhard Zumkeller, May 15 2006

Extensions

Name edited by Gus Wiseman, Oct 19 2019

Status

approved