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Numbers with a prime as third-smallest divisor.
15

%I #19 Jul 02 2022 05:58:46

%S 6,10,12,14,15,18,21,22,24,26,30,33,34,35,36,38,39,42,45,46,48,50,51,

%T 54,55,57,58,60,62,63,65,66,69,70,72,74,75,77,78,82,84,85,86,87,90,91,

%U 93,94,95,96,98,102,105,106,108,110,111,114,115,118,119,120,122,123,126

%N Numbers with a prime as third-smallest divisor.

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

%H Alois P. Heinz, <a href="/A119313/b119313.txt">Table of n, a(n) for n = 1..10000</a>

%e a(1) = A087134(3) = 6.

%e From _Gus Wiseman_, Oct 19 2019: (Start)

%e The sequence of terms together with their divisors begins:

%e 6: {1,2,3,6}

%e 10: {1,2,5,10}

%e 12: {1,2,3,4,6,12}

%e 14: {1,2,7,14}

%e 15: {1,3,5,15}

%e 18: {1,2,3,6,9,18}

%e 21: {1,3,7,21}

%e 22: {1,2,11,22}

%e 24: {1,2,3,4,6,8,12,24}

%e 26: {1,2,13,26}

%e 30: {1,2,3,5,6,10,15,30}

%e 33: {1,3,11,33}

%e 34: {1,2,17,34}

%e 35: {1,5,7,35}

%e 36: {1,2,3,4,6,9,12,18,36}

%e 38: {1,2,19,38}

%e 39: {1,3,13,39}

%e 42: {1,2,3,6,7,14,21,42}

%e 45: {1,3,5,9,15,45}

%e 46: {1,2,23,46}

%e (End)

%p q:= n-> (l-> nops(l)>2 and isprime(l[3]))(

%p sort([numtheory[divisors](n)[]])):

%p select(q, [$1..200])[]; # _Alois P. Heinz_, Oct 19 2019

%t Select[Range[100],Length[Divisors[#]]>2&&PrimeQ[Divisors[#][[3]]]&] (* _Gus Wiseman_, Oct 15 2019 *)

%t Select[Range[130], Length[f = FactorInteger[#]] > 1 && (f[[1, 2]] == 1 || f[[1, 1]]^2 > f[[2, 1]]) &] (* _Amiram Eldar_, Jul 02 2022 *)

%Y Complement of A119314.

%Y Subsequences: A006881, A000469, A008588.

%Y A subset of A002808 and A080257.

%Y Numbers whose third-largest divisor is prime are A328338.

%Y Second-smallest divisor is A020639.

%Y Third-smallest divisor is A292269.

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

%K nonn

%O 1,1

%A _Reinhard Zumkeller_, May 15 2006

%E Name edited by _Gus Wiseman_, Oct 19 2019