A217319 - OEIS

%I #42 Nov 22 2023 05:35:08

%S 4,8,12,20,24,28,36,40,44,52,56,60,64,68,72,76,84,88,92,100,104,108,

%T 116,120,124,128,132,136,140,148,152,156,164,168,172,180,184,188,192,

%U 196,200,204,212,216,220,228,232,236,244,248,252,260,264,268,276,280

%N Numbers with binary representation ending in 4*k+2 or 4*k+3 zeros.

%C Or numbers having infinitary divisor 4, or the same, having factor 4 in Fermi-Dirac representation as a product of distinct terms of A050376.

%C From _Peter Munn_, Aug 25 2020: (Start)

%C Compare the terms, as a set, with A145204\{0} (numbers having 3 as a Fermi-Dirac factor). The self-inverse function defined by A225546 maps the members of either one of these sets 1:1 onto the other set.

%C Numbers whose 4th-power-free part is divisible by 4.

%C (End)

%C Numbers k such that the exponent of the highest power of 4 dividing k, A235127(k), is odd. The asymptotic density of this sequence is 1/5. - _Amiram Eldar_, Sep 20 2020

%H Peter J. C. Moses, <a href="/A217319/b217319.txt">Table of n, a(n) for n = 1..10000</a>

%F Conjecture. For n>=1, a(n) = A171949(n+1).

%p isA007814 := proc(n)

%p     if modp( A007814(n),4) in {2,3} then

%p         true ;

%p     else

%p         false ;

%p     end if;

%p end proc:

%p for n from 1 to 1000 do

%p     if isA007814(n) then

%p         printf("%d,",n) ;

%p     end if;

%p end do: # _R. J. Mathar_, Nov 22 2023

%t okQ[n_] := (cnt = Count[ Split[ IntegerDigits[n, 2]] // Last, 0]; k0 = k /. ToRules@ Reduce[ (cnt == 2*k || cnt == 2*k+1), k, Integers]; OddQ[k0]); Select[ Range[312], okQ] (* _Jean-François Alcover_, Mar 18 2013 *)

%t Select[Map[# Boole[IntegerQ[(1/4 (1+#))]||IntegerQ[(1/4 (2+#))]&[Length[Last[Split[IntegerDigits[#,2]]]]]]&,Range[2,500,2]],#>0&]

%t Select[Range[280], OddQ @ IntegerExponent[#, 4] &] (* _Amiram Eldar_, Sep 20 2020 *)

%Y Cf. A050376, A171949, A235127.

%Y Related to A145204\{0} via A225546.

%K nonn,base

%O 1,1

%A _Vladimir Shevelev_, Mar 18 2013

%E Named edited by _David A. Corneth_, Sep 22 2020