A124169 -id:A124169 - cp's OEIS frontend

A055039 Numbers of the form 2^(2i+1)*(8j+7).

14, 30, 46, 56, 62, 78, 94, 110, 120, 126, 142, 158, 174, 184, 190, 206, 222, 224, 238, 248, 254, 270, 286, 302, 312, 318, 334, 350, 366, 376, 382, 398, 414, 430, 440, 446, 462, 478, 480, 494, 504, 510, 526, 542, 558, 568, 574, 590, 606, 622

Offset: 1

N. J. A. Sloane, Jun 01 2000

The numbers not of the form x^2+y^2+2z^2.
Numbers of the form 6*x^2 + 8*x^2*(2*y -1). (Steve Waterman).
These are the numbers not occurring as norms in the face-centered cubic lattice (cf. A004015).
Numbers whose base 4 representation ends in 3,2 followed by some number of zeros. - Franklin T. Adams-Watters, Dec 04 2006
Numbers k such that the k-th coefficient of eta(x)^4/eta(x^4) is 0 where eta is the Dedekind eta function. - Benoit Cloitre, Mar 15 2025
The asymptotic density of this sequence is 1/12. - Amiram Eldar, Mar 29 2025

			In base 4: 32, 132, 232, 320, 332, 1032, 1132, 1232, 1320, 1332, 2032, ...

T. D. Noe, Table of n, a(n) for n=1..10000
L. E. Dickson, Integers represented by positive ternary quadratic forms, Bull. Amer. Math. Soc. 33 (1927), 63-70.
L. J. Mordell, A new Waring's problem with squares of linear forms, Quart. J. Math., 1 (1930), 276-288 (see p. 283).
Steve Waterman, Missing numbers formula.

Equals twice A004215. Not the same as A044075 - see A124169.
Complement of A000401.
Cf. A004015.

Select[Range[650], Mod[# / 4^IntegerExponent[#, 4], 16] == 14 &] (* Amiram Eldar, Mar 29 2025 *)

from itertools import count, islice
def A055039_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:(m:=(~n&n-1).bit_length())&1 and (n>>m)&7==7,count(max(startvalue,1)))
A055039_list = list(islice(A055039_gen(),30)) # Chai Wah Wu, Jul 09 2022

def A055039(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(((x>>i)-7>>3)+1 for i in range(1,x.bit_length(),2))
    return bisection(f,n,n) # Chai Wah Wu, Feb 24 2025

A044075 Numbers k such that the string 3,2 occurs in the base-4 representation of k but not of k-1.

Original entry on oeis.org

14, 30, 46, 56, 62, 78, 94, 110, 120, 126, 142, 158, 174, 184, 190, 206, 222, 224, 248, 254, 270, 286, 302, 312, 318, 334, 350, 366, 376, 382, 398, 414, 430, 440, 446, 462, 478, 480, 504, 510, 526, 542, 558, 568, 574, 590, 606

Offset: 1

Clark Kimberling

Numbers whose base-4 representation ends in 3,2 followed by some number of zeros and includes no other 3,2. - Franklin T. Adams-Watters, Dec 04 2006
Not the same as A055039 - see A124169.
A 4-automatic set: membership is determined by comparing the base-4 representation of the number to the regular expression /[012]*(3+([01][012]*)?)*320*/. - Charles R Greathouse IV, Feb 11 2012 [corrected by Pontus von Brömssen, Jan 12 2019]
Alternatively, numbers whose base-4 representation is in the language generated by the regular expression /([012]|3*[01])*3+20*/. - Pontus von Brömssen, Jan 17 2019

R. J. Mathar, Table of n, a(n) for n = 1..10000
Index entries for 2-automatic sequences.

has32 := proc(n) local shft : shft := n : while shft > 0 do if shft mod 16 = 14 then RETURN(true) ; fi : shft := floor(shft/4) : od : RETURN(false) ; end: isA044075 := proc(n) if has32(n) and not has32(n-1) then return(true): else return(false) : fi : end: n := 1 : a := 1 : while n <= 10000 do while not isA044075(a) do a := a+1 : od : printf("%d %d ",n,a) : a := a+1 : n := n+1 : od : # R. J. Mathar, Dec 07 2006

Flatten[Position[Partition[Table[If[MemberQ[Partition[IntegerDigits[n, 4], 2, 1], {3, 2}], 1, 0], {n, 1000}], 2, 1], {0, 1}]] + 1 (* Vincenzo Librandi, Aug 19 2015 *)

cp's OEIS Frontend

A055039 Numbers of the form 2^(2i+1)*(8j+7).

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