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A119313 Numbers with a prime as third-smallest divisor.

%I A119313 #19 Jul 02 2022 05:58:46
%S A119313 6,10,12,14,15,18,21,22,24,26,30,33,34,35,36,38,39,42,45,46,48,50,51,
%T A119313 54,55,57,58,60,62,63,65,66,69,70,72,74,75,77,78,82,84,85,86,87,90,91,
%U A119313 93,94,95,96,98,102,105,106,108,110,111,114,115,118,119,120,122,123,126
%N A119313 Numbers with a prime as third-smallest divisor.
%C A119313 m is a term iff A001221(m) > 1 and (A067029(m) = 1 or A119288(m) < A020639(m)^2).
%H A119313 Alois P. Heinz, <a href="/A119313/b119313.txt">Table of n, a(n) for n = 1..10000</a>
%e A119313 a(1) = A087134(3) = 6.
%e A119313 From _Gus Wiseman_, Oct 19 2019: (Start)
%e A119313 The sequence of terms together with their divisors begins:
%e A119313     6: {1,2,3,6}
%e A119313    10: {1,2,5,10}
%e A119313    12: {1,2,3,4,6,12}
%e A119313    14: {1,2,7,14}
%e A119313    15: {1,3,5,15}
%e A119313    18: {1,2,3,6,9,18}
%e A119313    21: {1,3,7,21}
%e A119313    22: {1,2,11,22}
%e A119313    24: {1,2,3,4,6,8,12,24}
%e A119313    26: {1,2,13,26}
%e A119313    30: {1,2,3,5,6,10,15,30}
%e A119313    33: {1,3,11,33}
%e A119313    34: {1,2,17,34}
%e A119313    35: {1,5,7,35}
%e A119313    36: {1,2,3,4,6,9,12,18,36}
%e A119313    38: {1,2,19,38}
%e A119313    39: {1,3,13,39}
%e A119313    42: {1,2,3,6,7,14,21,42}
%e A119313    45: {1,3,5,9,15,45}
%e A119313    46: {1,2,23,46}
%e A119313 (End)
%p A119313 q:= n-> (l-> nops(l)>2 and isprime(l[3]))(
%p A119313          sort([numtheory[divisors](n)[]])):
%p A119313 select(q, [$1..200])[];  # _Alois P. Heinz_, Oct 19 2019
%t A119313 Select[Range[100],Length[Divisors[#]]>2&&PrimeQ[Divisors[#][[3]]]&] (* _Gus Wiseman_, Oct 15 2019 *)
%t A119313 Select[Range[130], Length[f = FactorInteger[#]] > 1 && (f[[1, 2]] == 1 || f[[1, 1]]^2 > f[[2, 1]]) &] (* _Amiram Eldar_, Jul 02 2022 *)
%Y A119313 Complement of A119314.
%Y A119313 Subsequences: A006881, A000469, A008588.
%Y A119313 A subset of A002808 and A080257.
%Y A119313 Numbers whose third-largest divisor is prime are A328338.
%Y A119313 Second-smallest divisor is A020639.
%Y A119313 Third-smallest divisor is A292269.
%Y A119313 Cf. A000005, A000040, A001221, A020639, A027750, A033676, A060775, A067029, A088725, A119288, A328189.
%K A119313 nonn
%O A119313 1,1
%A A119313 _Reinhard Zumkeller_, May 15 2006
%E A119313 Name edited by _Gus Wiseman_, Oct 19 2019