A055042 -id:A055042 - cp's OEIS frontend

A055045 Numbers of the form 4^i(8j+5).

5, 13, 20, 21, 29, 37, 45, 52, 53, 61, 69, 77, 80, 84, 85, 93, 101, 109, 116, 117, 125, 133, 141, 148, 149, 157, 165, 173, 180, 181, 189, 197, 205, 208, 212, 213, 221, 229, 237, 244, 245, 253, 261, 269, 276, 277, 285, 293, 301, 308, 309, 317, 320, 325, 333, 336, 340, 341

Offset: 1

N. J. A. Sloane, Jun 01 2000

Numbers not of the form x^2+2y^2+6z^2.

Reinhard Zumkeller, 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. See Theorem XI.
L. J. Mordell, A new Waring's problem with squares of linear forms, Quart. J. Math., 1 (1930), 276-288 (see p. 283).

Cf. A004215, A055042, A055046, A234000.

a055045 n = a055045_list !! (n-1)
a055045_list = filter ((== 5) . (flip mod 8) . f) [1..] where
   f x = if r == 0 then f x' else x  where (x', r) = divMod x 4
-- Reinhard Zumkeller, Jan 02 2014

A055045Q[k_] := Mod[k/4^IntegerExponent[k, 4], 8] == 5;
Select[Range[500], A055045Q] (* Paolo Xausa, Mar 20 2025 *)

is(n)=n/=4^valuation(n,4); n%8==5 \\ Charles R Greathouse IV and V. Raman, Dec 19 2013

from itertools import count, islice
def A055045_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:not (m:=(~n&n-1).bit_length())&1 and (n>>m)&7==5, count(max(startvalue, 1)))
A055045_list = list(islice(A055045_gen(), 30)) # Chai Wah Wu, Jul 09 2022

def A055045(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<<1))-5>>3)+1 for i in range(x.bit_length()>>1))
    return bisection(f,n,n) # Chai Wah Wu, Feb 14 2025

a(n) = 6n + O(log n). - Charles R Greathouse IV, Dec 19 2013

a(n) = A055042(n)/2. - Chai Wah Wu, Mar 19 2025

A203322 Numbers not of the form x^2+2y^3+3z^4 for nonnegative x, y, z.

Original entry on oeis.org

8, 10, 13, 15, 22, 24, 26, 29, 31, 33, 34, 37, 40, 42, 43, 45, 46, 47, 53, 56, 60, 62, 71, 72, 74, 76, 77, 78, 85, 87, 88, 91, 92, 94, 95, 96, 98, 101, 104, 107, 108, 109, 110, 115, 117, 120, 122, 125, 130, 133, 134, 136, 139, 141, 142, 143, 152, 155, 158, 159, 161, 162, 165, 168, 170, 173, 179, 181, 182, 184, 186, 187, 189

Offset: 1

N. J. A. Sloane, Jan 20 2012

nonn

Suggested by A055042.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A055042.

N:= 1000: # for terms <= N
E1:= {seq(3*z^4,z=0..floor((N/3)^(1/4)))}:
E2:= map(t -> seq(t+2*y^3,y=0..floor(((N-t)/2)^(1/3))), E1):
E3:= map(t -> seq(t+x^2,x=0..floor(sqrt(N-t))), E2):
sort(convert({$1..N} minus E3,list)); # Robert Israel, Aug 17 2020

A366091 a(n) is the number of ways to write n = i^2 + 2j^2 + 3k^2 with i,j,k >= 0.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 1, 1, 3, 0, 2, 4, 1, 2, 2, 2, 1, 3, 2, 2, 4, 2, 1, 2, 2, 0, 4, 3, 2, 5, 2, 1, 3, 2, 2, 7, 2, 2, 5, 0, 2, 0, 2, 4, 4, 3, 1, 4, 3, 3, 5, 3, 2, 7, 1, 2, 6, 0, 3, 6, 2, 2, 4, 2, 2, 6, 3, 2, 4, 3, 3, 3, 2, 0, 7, 5, 2, 6, 3, 2, 8, 2, 2, 11, 2, 5, 2, 2, 3, 0, 4, 3, 7, 3, 2, 2, 3, 3

Offset: 0

Robert Israel, Sep 28 2023

nonn

			a(9) = 3 because 9 = 3^2 + 2*0^2 + 3*0^2 = 1^2 + 2*2^2 + 3*0^2 = 2^2 + 2*1^2 + 3*1^2.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A028594 (allows any integer i,j,k), A055042 (a(n) = 0)

g:= (1+JacobiTheta3(0,z))*(1+JacobiTheta3(0,z^2))*(1+JacobiTheta3(0,z^3))/8:
S:= series(g,z,101):
seq(coeff(S,z,j),j=0..100);

from itertools import count
from sympy.ntheory.primetest import is_square
def A366091(n):
    c = 0
    for k in count(0):
        if (a:=3*k**2)>n:
            break
        for j in count(0):
            if (b:=a+(j**2<<1))>n:
                break
            if is_square(n-b):
                c += 1
    return c # Chai Wah Wu, Sep 29 2023

G.f. (1 + theta_3(0,z)) * (1 + theta_3(0,z^2)) * (1 + theta_3(0,z^3))/8 where theta_3 is a Jacobi theta function.

A370267 Numbers with an even number of prime factors not of the form 8m+-1 (counting repetitions).

Original entry on oeis.org

1, 4, 6, 7, 9, 10, 15, 16, 17, 22, 23, 24, 25, 26, 28, 31, 33, 36, 38, 39, 40, 41, 42, 47, 49, 54, 55, 57, 58, 60, 63, 64, 65, 68, 70, 71, 73, 74, 79, 81, 86, 87, 88, 89, 90, 92, 95, 96, 97, 100, 102, 103, 104, 105, 106, 111, 112, 113, 118, 119, 121, 122, 124, 127, 129

Offset: 1

Peter Munn, Feb 13 2024

Construction by subgroup generation: (Start)
The set of numbers congruent to 1 modulo 8 (A017077) contains all the odd squares and generates a subgroup of the positive rational numbers (under multiplication) that contains no additional integers. The subgroup has an infinite number of cosets. The rest of the construction process extends the subgroup, reducing the number of cosets to 2, by choosing additional generators that are semiprime.
First we extend the subgroup to include all nonzero integer squares. As we already have the odd squares, we need only add 4, the square of the smallest prime, as a generator. The extended subgroup has only 8 cosets and its integer members are listed in A234000. To achieve a subgroup with 2 cosets we now add squarefree semiprime generators. The 2 smallest, 6 and 10, suffice.
The resulting subgroup has this sequence's terms as its integer members.
(End)
The equivalent process starting with numbers congruent to 1 modulo 3 (or 1 modulo 6) produces A189715. If we take its intersection with this sequence we get A370268, which starts with the first 72 nonzero numbers of the form x^2 + 6y^2 (see A002481). Similarly, if we start with numbers congruent to 1 modulo 5 (or 1 modulo 10) and take the resulting set's intersection with this sequence we get a set starting with the first 32 nonzero numbers of the form x^2 - 10y^2 (see A242664).
The construction process leads to a number of properties:
- The sequence is closed under multiplication and all integer ratios between terms are in the sequence.
- The sequence and its complement have the property that the terms of one can be generated by halving the even terms of the other. Each has asymptotic density 1/2.
Numbers whose squarefree part is congruent to {1,7} mod 8 or {6,10} mod 16.

			7 is prime, so 7 is its only prime factor, which has the form 8m-1. So 7 has an even number (zero) of prime factors not of the form 8m+-1, and therefore is in the sequence. In terms of the subgroup generators described at the start of the comments, (13*8+1) * 4 / (6*10) = 105 * 4/60 = 7.
110 = 2 * 5 * 11, so it has 3 prime factors and all 3 do not have the form 8m+-1. 3 is odd, so 110 is not in the sequence.

Disjoint union of A004215, A055042, A055043 and A234000.
See the comments for the relationships with A002481, A017077, A189715, A242664, A370268.
Cf. A042999 (primes), A059897.

isok(k) = {c = core(k); c%8 == 1 || c%8 == 7 || c%16 == 6 || c%16 == 10}

def A370267(n):
    def f(x): return n+x-sum(((y:=x>>(i<<1))-7>>3)+(y-1>>3)+2 for i in range((x.bit_length()>>1)+1))-sum(((z:=x>>(i<<1)+1)-5>>3)+(z-3>>3)+2 for i in range(x.bit_length()-1>>1))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Mar 19 2025

{a(n) : n >= 1} = {A059897(i,j) : i in A234000, j in {1, 6, 10, 15}}.

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A055045 Numbers of the form 4^i*(8*j+5).

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A203322 Numbers not of the form x^2+2*y^3+3*z^4 for nonnegative x, y, z.

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A366091 a(n) is the number of ways to write n = i^2 + 2*j^2 + 3*k^2 with i,j,k >= 0.

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Programs

Maple

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A370267 Numbers with an even number of prime factors not of the form 8m+-1 (counting repetitions).

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Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Python

Formula

A055045 Numbers of the form 4^i(8j+5).

A203322 Numbers not of the form x^2+2y^3+3z^4 for nonnegative x, y, z.

A366091 a(n) is the number of ways to write n = i^2 + 2j^2 + 3k^2 with i,j,k >= 0.