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A348005 Positive even integers with an even number of even divisors.

%I A348005 #47 Aug 04 2022 15:04:23
%S A348005 4,6,10,12,14,16,20,22,24,26,28,30,34,36,38,40,42,44,46,48,52,54,56,
%T A348005 58,60,62,64,66,68,70,74,76,78,80,82,84,86,88,90,92,94,96,100,102,104,
%U A348005 106,108,110,112,114,116,118,120,122,124,126,130,132,134,136,138,140,142,144,146,148,150,152,154,156
%N A348005 Positive even integers with an even number of even divisors.
%C A348005 These terms are exactly the even numbers in A183300.
%C A348005 Complement of A001105 relative to the positive even integers (A005843 \ {0}).
%C A348005 Note that odd integers with an odd number of odd divisors are the odd squares (A016754).
%F A348005 a(n) = 2*A000037(n).
%e A348005 The divisors of 14 are {1, 2, 7, 14}, two of them: 2 and 14 are even, hence 14 is a term.
%e A348005 The divisors of 16 are {1, 2, 4, 8, 16}, four of them: 2, 4, 8 and 16 are even, hence 16 is another term.
%p A348005 filter:= q -> irem(q, 2) = 0 and sqrt(q/2) <> floor(sqrt(q/2)) : select(filter, [$1..156]);
%t A348005 m = 9; 2 * Complement[Range[m^2], Range[m]^2] (* _Amiram Eldar_, Oct 02 2021 *)
%o A348005 (PARI) isok(k) = !(k % 2) && !(sumdiv(k, d, !(d % 2)) % 2); \\ _Michel Marcus_, Oct 05 2021
%o A348005 (Python)
%o A348005 from math import isqrt
%o A348005 def A348005(n): return n+(m:=isqrt(n))+int(n-m*(m+1)>=1)<<1 # _Chai Wah Wu_, Aug 04 2022
%Y A348005 Cf. A000037, A001105, A005843, A016754.
%Y A348005 Equals A183300 \ A005408.
%Y A348005 Intersection of A005843 and A183300.
%Y A348005      -------------------------------------------------------------------------
%Y A348005     |     Integers with    |  an even number of ...   |  an odd number of ... |
%Y A348005      -------------------------------------------------------------------------
%Y A348005     |  ... even divisors   |         A183300          |        A001105        |
%Y A348005     |  ... odd divisors    |         A028983          |        A028982        |
%Y A348005      -------------------------------------------------------------------------
%K A348005 nonn
%O A348005 1,1
%A A348005 _Bernard Schott_, Oct 02 2021