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Positive even integers with an even number of even divisors.
2

%I #47 Aug 04 2022 15:04:23

%S 4,6,10,12,14,16,20,22,24,26,28,30,34,36,38,40,42,44,46,48,52,54,56,

%T 58,60,62,64,66,68,70,74,76,78,80,82,84,86,88,90,92,94,96,100,102,104,

%U 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 Positive even integers with an even number of even divisors.

%C These terms are exactly the even numbers in A183300.

%C Complement of A001105 relative to the positive even integers (A005843 \ {0}).

%C Note that odd integers with an odd number of odd divisors are the odd squares (A016754).

%F a(n) = 2*A000037(n).

%e The divisors of 14 are {1, 2, 7, 14}, two of them: 2 and 14 are even, hence 14 is a term.

%e 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 filter:= q -> irem(q, 2) = 0 and sqrt(q/2) <> floor(sqrt(q/2)) : select(filter, [$1..156]);

%t m = 9; 2 * Complement[Range[m^2], Range[m]^2] (* _Amiram Eldar_, Oct 02 2021 *)

%o (PARI) isok(k) = !(k % 2) && !(sumdiv(k, d, !(d % 2)) % 2); \\ _Michel Marcus_, Oct 05 2021

%o (Python)

%o from math import isqrt

%o def A348005(n): return n+(m:=isqrt(n))+int(n-m*(m+1)>=1)<<1 # _Chai Wah Wu_, Aug 04 2022

%Y Cf. A000037, A001105, A005843, A016754.

%Y Equals A183300 \ A005408.

%Y Intersection of A005843 and A183300.

%Y -------------------------------------------------------------------------

%Y | Integers with | an even number of ... | an odd number of ... |

%Y -------------------------------------------------------------------------

%Y | ... even divisors | A183300 | A001105 |

%Y | ... odd divisors | A028983 | A028982 |

%Y -------------------------------------------------------------------------

%K nonn

%O 1,1

%A _Bernard Schott_, Oct 02 2021