A072400 -id:A072400 - cp's OEIS frontend

A286366 Compound filter: a(n) = 2*A286365(n) + floor(A072400(n)/4).

4, 6, 8, 4, 13, 11, 9, 6, 28, 14, 8, 8, 13, 11, 21, 4, 12, 30, 8, 13, 65, 11, 9, 11, 40, 14, 116, 9, 13, 23, 9, 6, 64, 14, 20, 28, 13, 11, 21, 14, 12, 66, 8, 8, 49, 11, 9, 8, 28, 42, 20, 13, 13, 119, 21, 11, 64, 14, 8, 21, 13, 11, 269, 4, 84, 66, 8, 12, 65, 23, 9, 30, 12, 14, 56, 8, 65, 23, 9, 13, 484, 14, 8, 65, 85, 11, 21, 11, 12, 50, 20, 9, 65, 11, 21, 11

Offset: 1

Antti Karttunen, May 08 2017

nonn

Each term of this sequence contains, in addition to the information contained in A286365 (which packs the values of A286361(n) and A286363(n) and parity of the exponent of the highest power of 2 dividing n) also the bit-2 of A072400(n) (its third least significant bit), which is here stored as the least significant bit of a(n). Note that the whole A072400(n) can be recovered based on the other information contained in a(n). Together all this information is enough - by Lagrange's "Four Squares theorem" - to determine what is the least number of squares that add up to n. Thus it follows that for all i, j: a(i) = a(j) => A002828(i) = A002828(j).

A286369 is similar, but without the parity of the 2-adic value present.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000035, A002828, A007814, A072400, A286361, A286363, A286364, A286365, A286368, A286369, A286372, A286373.

from sympy import factorint
from operator import mul
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a046523(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1
def A(n, k):
    f = factorint(n)
    return 1 if n == 1 else reduce(mul, [1 if i%4==k else i**f[i] for i in f])
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def a286364(n): return T(a046523(n/A(n, 1)), a046523(n/A(n, 3)))
def a007814(n): return 1 + bin(n - 1)[2:].count("1") - bin(n)[2:].count("1")
def a286365(n): return 2*a286364(n) + a007814(n)%2
def a072400(n): return int(str(int(''.join(map(str, digits(n, 4)[1:]))[::-1]))[::-1], 4)%8
def a(n): return 2*a286365(n) + int(a072400(n)/4) # Indranil Ghosh, May 09 2017

(define (A286366 n) (+ (* 2 (A286365 n)) (floor->exact (/ (A072400 n) 4))))

a(n) = 2*A286365(n) + floor(A072400(n)/4).

A286369 Compound filter: a(n) = 2*A286364(n) + floor(A072400(n)/4).

Original entry on oeis.org

2, 2, 4, 2, 7, 5, 5, 2, 14, 6, 4, 4, 7, 5, 11, 2, 6, 14, 4, 7, 33, 5, 5, 5, 20, 6, 58, 5, 7, 11, 5, 2, 32, 6, 10, 14, 7, 5, 11, 6, 6, 32, 4, 4, 25, 5, 5, 4, 14, 20, 10, 7, 7, 59, 11, 5, 32, 6, 4, 11, 7, 5, 135, 2, 42, 32, 4, 6, 33, 11, 5, 14, 6, 6, 28, 4, 33, 11, 5, 7, 242, 6, 4, 33, 43, 5, 11, 5, 6, 24, 10, 5, 33, 5, 11, 5, 6, 14, 134, 20, 7, 11, 5, 6, 46, 6

Offset: 1

Antti Karttunen, May 09 2017

nonn

This sequence contains, in addition to the information contained in A286364 (which packs the values of A286361(n) and A286363(n) to a single value with the pairing function A000027) also the bit-2 of A072400(n) (its third least significant bit), which is here stored as the least significant bit of a(n). In contrast to A286366, the parity of the highest power of 2 dividing n is not stored.
Thus we have (among other such identities) the following two identities related to equivalence class partitioning:
For all odd i, odd j: a(i) = a(j) <=> A286366(i) = A286366(j).
For all odd i, odd j: a(i) = a(j) => A010877(i) = A010877(j). [On odd numbers the information contained in a(n) is sufficient to determine the value of n modulo 8, one of the 1, 3, 5 or 7.]

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A072400, A286364, A286366, A286370, A286371.

from sympy.ntheory.factor_ import digits
from sympy import factorint
from operator import mul
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a046523(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1
def A(n, k):
    f = factorint(n)
    return 1 if n == 1 else reduce(mul, [1 if i%4==k else i**f[i] for i in f])
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def a286364(n): return T(a046523(n/A(n, 1)), a046523(n/A(n, 3)))
def a072400(n): return int(str(int(''.join(map(str, digits(n, 4)[1:]))[::-1]))[::-1], 4)%8
def a(n): return 2*a286364(n) + int(a072400(n)/4) # Indranil Ghosh, May 09 2017

(define (A286369 n) (+ (* 2 (A286364 n)) (floor->exact (/ (A072400 n) 4))))

a(n) = 2*A286364(n) + floor(A072400(n)/4).

A000378 Sums of three squares: numbers of the form x^2 + y^2 + z^2.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 29, 30, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83

Offset: 1

N. J. A. Sloane

nonn

An equivalent definition: numbers of the form x^2 + y^2 + z^2 with x,y,z >= 0.
Bourgain studies "the spatial distribution of the representation of a large integer as a sum of three squares, on the small and critical scale as well as their electrostatic energy. The main results announced give strong evidence to the thesis that the solutions behave randomly. This is in sharp contrast to what happens with sums of two or four or more square." Sums of two nonzero squares are A000404. - Jonathan Vos Post, Apr 03 2012
The multiplicities for a(n) (if 0 <= x <= y <= z) are given as A000164(a(n)), n >= 1. Compare with A005875(a(n)) for integer x, y and z, and order taken into account. - Wolfdieter Lang, Apr 08 2013
a(n)^k is a member of this sequence for any k > 1. - Boris Putievskiy, May 05 2013
The selection rule for the planes with Miller indices (hkl) to undergo X-ray diffraction in a simple cubic lattice is h^2+k^2+l^2 = N where N is a term of this sequence. See A004014 for f.c.c. lattice. - Mohammed Yaseen, Nov 06 2022

			a(1) = 0 = 0^2 + 0^2 + 0^2. A005875(0) = 1 = A000164(0).
a(9) = 9 = 0^2 + 0^2 + 3^2 =  1^2 +  2^2 + 2^2. A000164(9) = 2. A000164(9) = 30 = 2*3 + 8*3 (counting signs and order). - _Wolfdieter Lang_, Apr 08 2013

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 107.
E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 37.
R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section C20.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 311.

T. D. Noe, Table of n, a(n) for n = 1..10000
Jean Bourgain, Peter Sarnak and Zeév Rudnick, Local statistics of lattice points on the sphere, arXiv:1204.0134 [math.NT], 2012-2015.
J. W. Cogdell, On sums of three squares, Journal de théorie des nombres de Bordeaux, 15 no. 1 (2003), p. 33-44.
L. E. Dickson, Integers represented by positive ternary quadratic forms, Bull. Amer. Math. Soc. 33 (1927), 63-70. See Theorem I.
Eric T. Mortenson, A Kronecker-type identity and the representations of a number as a sum of three squares, arXiv:1702.01627 [math.NT], 2017.
P. Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 91. [?Broken link]
P. Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 91.
Eric Weisstein's World of Mathematics, Square Number
Index entries for sequences related to sums of squares

Union of A000290, A000404 and A000408 (common elements).
Union of A000290, A000415 and A000419 (disjunct sets).
Complement of A004215.
Cf. A005875 (number of representations if x, y and z are integers).
Cf. A047449, A034043, A034044, A034045, A034046, A034047, A065883, A072400, A072401, A071374, A002828, A001481, A125084, A000164.

isA000378 := proc(n) # return true or false depending on n being in the list
    local x,y ;
    for x from 0 do
        if 3*x^2 > n then
            return false;
        end if;
        for y from x do
            if x^2+2*y^2 > n then
                break;
            else
                if issqr(n-x^2-y^2) then
                    return true;
                end if;
            end if;
        end do:
    end do:
end proc:
A000378 := proc(n) # generate A000378(n)
    option remember;
    local a;
    if n = 1 then
        0;
    else
        for a from procname(n-1)+1 do
            if isA000378(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A000378(n),n=1..100) ; # R. J. Mathar, Sep 09 2015

okQ[n_] := If[EvenQ[k = IntegerExponent[n, 2]], m = n/2^k; Mod[m, 8] != 7, True]; Select[Range[0, 100], okQ] (* Jean-François Alcover, Feb 08 2016, adapted from PARI *)

isA000378(n)=my(k=valuation(n, 2)); if(k%2==0, n>>=k; n%8!=7, 1)

list(lim)=my(v=List(),k,t); for(x=0,sqrtint(lim\=1), for(y=0, min(sqrtint(lim-x^2),x), k=x^2+y^2; for(z=0,min(sqrtint(lim-k), y), listput(v,k+z^2)))); Set(v) \\ Charles R Greathouse IV, Sep 14 2015

def valuation(n, b):
    v = 0
    while n > 1 and n%b == 0: n //= b; v += 1
    return v
def ok(n): return n//4**valuation(n, 4)%8 != 7
print(list(filter(ok, range(84)))) # Michael S. Branicky, Jul 15 2021

from itertools import count, islice
def A000378_gen(): # generator of terms
    return filter(lambda n:n>>2*(bin(n)[:1:-1].index('1')//2) & 7 < 7, count(1))
A000378_list = list(islice(A000378_gen(),30)) # Chai Wah Wu, Jun 27 2022

def A000378(n):
    def f(x): return n-1+sum(((x>>(i<<1))-7>>3)+1 for i in range(x.bit_length()>>1))
    m, k = n-1, f(n-1)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Feb 14 2025

Legendre: a nonnegative integer is a sum of three squares iff it is not of the form 4^k m with m == 7 (mod 8).
n^(2k+1) is in the sequence iff n is in the sequence. - Ray Chandler, Feb 03 2009
Complement of A004215; complement of A000302(i)*A004771(j), i,j>=0. - Boris Putievskiy, May 05 2013
a(n) = 6n/5 + O(log n). - Charles R Greathouse IV, Mar 14 2014

More terms from Ray Chandler, Sep 05 2004

A072401 1 iff n is of the form 4^m*(8k+7).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0

Offset: 0

Reinhard Zumkeller, Jun 16 2002

Characteristic function of A004215, indicating numbers not the sum of 3 integer squares.

a(n) + 1 is the smallest positive number such that (a(n) + 1) * n is the sum of three squares. - Peter Schorn, Jul 18 2023

Jean-Paul Allouche and Jeffrey Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197; preprint. See Example 20.
Eric Weisstein's World of Mathematics, Square Numbers.
Index entries for sequences related to sums of squares.
Index entries for characteristic functions.

Cf. A004215, A057427, A064873, A072400.

Cf. A071374.

A072400[n_] := Mod[If[Mod[n, 4] == 0, n/4^IntegerExponent[n, 4], n], 8];
a[n_] := 1 - Sign[7 - A072400[n]];
Table[a[n], {n, 0, 96}] (* Jean-François Alcover, Dec 13 2021 *)

a(n) = if(n, (n >> (2*valuation(n, 4))) % 8 == 7, 0); \\ Amiram Eldar, May 15 2025

def A072401(n): return ((m:=(~n&n-1).bit_length())&1^1)&int((n>>m)&7==7) # Chai Wah Wu, Aug 01 2023

a(n) = 1 - A057427(7 - A072400(n)).
a(A004215(k)) = 1 for k>0.
a(n) = A057427(A064873(n)).
For n<112: a(n) = A064873(n), but A064873(112) = 2, as also a(112 - 1) = 1.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/6. - Amiram Eldar, May 15 2025

cp's OEIS Frontend

A286366 Compound filter: a(n) = 2*A286365(n) + floor(A072400(n)/4).

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A286369 Compound filter: a(n) = 2*A286364(n) + floor(A072400(n)/4).

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A000378 Sums of three squares: numbers of the form x^2 + y^2 + z^2.

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A072401 1 iff n is of the form 4^m*(8k+7).

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Mathematica

PARI

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