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A217319 Numbers with binary representation ending in 4*k+2 or 4*k+3 zeros.

%I A217319 #49 May 25 2025 10:08:17
%S A217319 4,8,12,20,24,28,36,40,44,52,56,60,64,68,72,76,84,88,92,100,104,108,
%T A217319 116,120,124,128,132,136,140,148,152,156,164,168,172,180,184,188,192,
%U A217319 196,200,204,212,216,220,228,232,236,244,248,252,260,264,268,276,280
%N A217319 Numbers with binary representation ending in 4*k+2 or 4*k+3 zeros.
%C A217319 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 A217319 From _Peter Munn_, Aug 25 2020: (Start)
%C A217319 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 A217319 Numbers whose 4th-power-free part is divisible by 4.
%C A217319 (End)
%C A217319 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 A217319 Peter J. C. Moses, <a href="/A217319/b217319.txt">Table of n, a(n) for n = 1..10000</a>
%F A217319 Conjecture. For n>=1, a(n) = A171949(n+1).
%p A217319 isA007814 := proc(n)
%p A217319     if modp( A007814(n),4) in {2,3} then
%p A217319         true ;
%p A217319     else
%p A217319         false ;
%p A217319     end if;
%p A217319 end proc:
%p A217319 for n from 1 to 1000 do
%p A217319     if isA007814(n) then
%p A217319         printf("%d,",n) ;
%p A217319     end if;
%p A217319 end do: # _R. J. Mathar_, Nov 22 2023
%t A217319 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 A217319 Select[Map[# Boole[IntegerQ[(1/4 (1+#))]||IntegerQ[(1/4 (2+#))]&[Length[Last[Split[IntegerDigits[#,2]]]]]]&,Range[2,500,2]],#>0&]
%t A217319 Select[Range[280], OddQ @ IntegerExponent[#, 4] &] (* _Amiram Eldar_, Sep 20 2020 *)
%o A217319 (Python)
%o A217319 def A217319(n):
%o A217319     def f(x): return n+x-sum((k:=x>>(m<<1))-(k>>2) for m in range(0,(x.bit_length()+1>>1),2))
%o A217319     m, k = n, f(n)
%o A217319     while m != k: m, k = k, f(k)
%o A217319     return m<<2 # _Chai Wah Wu_, May 25 2025
%Y A217319 Cf. A050376, A171949, A235127.
%Y A217319 Related to A145204\{0} via A225546.
%K A217319 nonn,base
%O A217319 1,1
%A A217319 _Vladimir Shevelev_, Mar 18 2013
%E A217319 Named edited by _David A. Corneth_, Sep 22 2020