A306199 - OEIS

%I #29 Mar 25 2021 04:56:40

%S 4,8,16,20,28,32,40,44,48,52,56,64,68,76,80,88,92,96,100,104,112,116,

%T 124,128,136,140,148,152,160,164,172,176,184,188,192,196,200,208,212,

%U 220,224,232,236,240

%N Numbers k having the property that tau(4*k) < tau(3*k) where tau = A000005.

%C All terms are divisible by 4.

%C A092259 (numbers congruent to {4,8} (mod 12)) is a subset.

%C Sequence also includes all numbers of the form 48*k where k is congruent to {1,2} (mod 3) (A001651).

%C Additional entries of the form 48k, where k is divisible by three have k values of 12*{1,2,4,5,7,8,10,11,12,13,14,16,17,19,20,22,23,24,...}

%C From _Robert Israel_, Jan 29 2019: (Start)

%C Numbers k such that A007814(k)- 2*A007949(k) >= 2.

%C Sequence is closed under multiplication. (End)

%C The asymptotic density of this sequence is 2/11. - _Amiram Eldar_, Mar 25 2021

%H Robert Israel, <a href="/A306199/b306199.txt">Table of n, a(n) for n = 1..10000</a>

%e tau(4*20) = 10, tau(3*20)=12. So 20 is in the sequence.

%p with(numtheory): for n from 1 to 300 do if tau(4*n) < tau(3*n) then print(n) fi od

%t Select[Range[4, 240, 4], #1 < #2 & @@ DivisorSigma[0, # {4, 3}] &] (* _Michael De Vlieger_, Jan 29 2019 *)

%t Select[Range[240], IntegerExponent[#, 2] - 2 * IntegerExponent[#, 3] >= 2 &] (* _Amiram Eldar_, Mar 25 2021 *)

%Y Cf. A000005, A001651, A007814, A007949, A092259.

%K nonn

%O 1,1

%A _Gary Detlefs_, Jan 28 2019