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

4, 8, 12, 20, 24, 28, 36, 40, 44, 52, 56, 60, 64, 68, 72, 76, 84, 88, 92, 100, 104, 108, 116, 120, 124, 128, 132, 136, 140, 148, 152, 156, 164, 168, 172, 180, 184, 188, 192, 196, 200, 204, 212, 216, 220, 228, 232, 236, 244, 248, 252, 260, 264, 268, 276, 280

Offset: 1

Author

Vladimir Shevelev, Mar 18 2013

Keywords

• nonn
• base

Comments

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

From Peter Munn, Aug 25 2020: (Start)

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.

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

(End)

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

Links

• Peter J. C. Moses, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A050376, A171949, A235127.

Related to A145204\{0} via A225546.

Programs

Maple

isA007814 := proc(n)
    if modp( A007814(n),4) in {2,3} then
        true ;
    else
        false ;
    end if;
end proc:
for n from 1 to 1000 do
    if isA007814(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Nov 22 2023

Mathematica

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 *)
Select[Map[# Boole[IntegerQ[(1/4 (1+#))]||IntegerQ[(1/4 (2+#))]&[Length[Last[Split[IntegerDigits[#,2]]]]]]&,Range[2,500,2]],#>0&]
Select[Range[280], OddQ @ IntegerExponent[#, 4] &] (* Amiram Eldar, Sep 20 2020 *)

Python

def A217319(n):
    def f(x): return n+x-sum((k:=x>>(m<<1))-(k>>2) for m in range(0,(x.bit_length()+1>>1),2))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m<<2 # Chai Wah Wu, May 25 2025

Formula

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

Extensions

Named edited by David A. Corneth, Sep 22 2020