A134738 -id:A134738 - cp's OEIS frontend

A004215 Numbers that are the sum of 4 but no fewer nonzero squares.

7, 15, 23, 28, 31, 39, 47, 55, 60, 63, 71, 79, 87, 92, 95, 103, 111, 112, 119, 124, 127, 135, 143, 151, 156, 159, 167, 175, 183, 188, 191, 199, 207, 215, 220, 223, 231, 239, 240, 247, 252, 255, 263, 271, 279, 284, 287, 295, 303, 311, 316, 319, 327, 335, 343

Offset: 1

N. J. A. Sloane and J. H. Conway

Lagrange's theorem tells us that each positive integer can be written as a sum of four squares.
If n is in the sequence and k is an odd positive integer then n^k is in the sequence because n^k is of the form 4^i(8j+7). - Farideh Firoozbakht, Nov 23 2006
Numbers whose cubes do not have a partition as a sum of 3 squares. a(n)^3 = A134738(n). - Artur Jasinski, Nov 07 2007
A002828(a(n)) = 4; A025427(a(n)) > 0. - Reinhard Zumkeller, Feb 26 2015
There are infinitely many adjacent pairs (for example, 128n + 111 and 128n + 112 for any n), but never a triple of consecutive integers. - Ivan Neretin, Aug 17 2017
These numbers are called "forbidden numbers" in crystallography: for a cubic crystal, no reflection with index hkl such that h^2 + k^2 + l^2 = a(n) appears in the crystal's diffraction pattern. - A. Timothy Royappa, Aug 11 2021

			15 is in the sequence because it is the sum of four squares, namely, 3^2 + 2^2 + 1^2 + 1^2, and it can't be expressed as the sum of fewer squares.
16 is not in the sequence, because, although it can be expressed as 2^2 + 2^2 + 2^2 + 2^2, it can also be expressed as 4^2.

L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 261.
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 12.
E. Poznanski, 1901. Pierwiastki pierwotne liczb pierwszych. Warszawa, pp. 1-63.
W. Sierpiński, 1925. Teorja Liczb. pp. 1-410 (p. 125).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, entry 4181.

T. D. Noe, Table of n, a(n) for n = 1..10000
David S. Bettes, Letter to N. J. A. Sloane, Nov 05 1976
Richard T. Bumby, Sums Of Four Squares
International Union of Crystallography, Cubic structures.
Shuo Li, The characteristic sequence of the integers that are the sum of two squares is not morphic, arXiv:2404.08822 [math.NT], 2024.
Louis J. Mordell, A new Waring's problem with squares of linear forms, Quart. J. Math., 1 (1930), 276-288 (see p. 283).
Saburô Uchiyama, A five-square theorem, Publ. Res. Math. Sci., Vol 13, Number 1 (1977), 301-305.
Steve Waterman, Missing numbers formula
Eric Weisstein's World of Mathematics, Square Number
Wikipedia, Lagrange's four-square theorem.
Index entries for sequences related to sums of squares

Complement of A000378.
Cf. A002828, A055039, A072401, A125084, A134738, A134739, A055045, A055046, A234000.
Cf. A000118 (ways to write n as sum of 4 squares), A025427.

a004215 n = a004215_list !! (n-1)
a004215_list = filter ((== 4) . a002828) [1..]
-- Reinhard Zumkeller, Feb 26 2015

N:= 1000: # to get all terms <= N
{seq(seq(4^i * (8*j + 7), j = 0 .. floor((N/4^i - 7)/8)), i = 0 .. floor(log[4](N)))}; # Robert Israel, Sep 02 2014

Sort[Flatten[Table[4^i(8j + 7), {i, 0, 2}, {j, 0, 42}]]] (* Alonso del Arte, Jul 05 2005 *)
Select[Range[120], Mod[ #/4^IntegerExponent[ #, 4], 8] == 7 &] (* Ant King, Oct 14 2010 *)

isA004215(n)={ local(fouri,j) ; fouri=1 ; while( n >=7*fouri, if( n % fouri ==0, j= n/fouri -7 ; if( j % 8 ==0, return(1) ) ; ) ; fouri *= 4 ; ) ; return(0) ; } { for(n=1,400, if(isA004215(n), print1(n,",") ; ) ; ) ; } \\ R. J. Mathar, Nov 22 2006

isA004215(n)= n\4^valuation(n,4)%8==7 \\ M. F. Hasler, Mar 18 2011

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 # after M. F. Hasler
print(list(filter(ok, range(344)))) # Michael S. Branicky, Jul 15 2021

from itertools import count, islice
def A004215_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:not (m:=(~n&n-1).bit_length())&1 and (n>>m)&7==7,count(max(startvalue,1)))
A004215_list = list(islice(A004215_gen(),30)) # Chai Wah Wu, Jul 09 2022

def A004215(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))-7>>3)+1 for i in range(x.bit_length()>>1))
    return bisection(f,n,n) # Chai Wah Wu, Feb 14 2025

a(n) = A055039(n)/2. - Ray Chandler, Jan 30 2009
Numbers of the form 4^i*(8*j+7), i >= 0, j >= 0. [A.-M. Legendre & C. F. Gauss]
Products of the form A000302(i)*A004771(j), i, j >= 0. - R. J. Mathar, Nov 29 2006
a(n) = 6*n + O(log(n)). - Charles R Greathouse IV, Dec 19 2013
Conjecture: The number of terms < 2^n is A023105(n) - 2. - Tilman Neumann, Sep 20 2020

More terms from Arlin Anderson (starship1(AT)gmail.com)

Additional comments from Jud McCranie, Mar 19 2000

A134739 Cubes of (positive numbers that are not the sum of three nonzero squares), that is, the terms of A004214, cubed.

Original entry on oeis.org

1, 8, 64, 125, 343, 512, 1000, 2197, 3375, 4096, 8000, 12167, 15625, 21952, 29791, 32768, 50653, 59319, 64000, 103823, 140608, 166375, 195112, 216000, 250047, 262144, 357911, 493039, 512000, 614125, 658503, 778688, 857375, 1000000

Offset: 1

Artur Jasinski, Nov 07 2007

nonn

This sequence was inspired by e-mail from Ray Chandler, Nov 07 2007.
Original name was: Cubes which are not the sum of three nonzero squares. That definition would not include 125 = 5^2 + 6^2 + 8^2. - Robert Israel, Jan 12 2016
For "(cubes of positive numbers) that are not the sum of three nonzero squares", that is, the cubes in A004214, see A267189. - N. J. A. Sloane, Jan 18 2016

			8 is in the sequence because it is not possible to express 2 as a sum of three nonzero squares and 2^3 = 8.
27 is not in the sequence because 3 = 1^2 + 1^2 + 1^2.

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

Cf. A004214, A134738, A267189.

N:= 1000: # to get all terms <= N^3
A004214:= {$1..N} minus {seq(seq(seq(a^2 + b^2 + c^2, c = b .. floor(sqrt(N-a^2-b^2))), b = a .. floor(sqrt(N-a^2))),a=1..floor(sqrt(N/2)))}:
map(`^`,sort(convert(A004214,list)), 3); # Robert Israel, Jan 12 2016

searchMax = 16; Flatten[Position[Take[Rest[CoefficientList[Sum[x^(i^2), {i, searchMax}]^3, x]], searchMax^2], 0]]^3 (* Based on Ray Chandler's program for A004214, Alonso del Arte, Jan 12 2016 *)

is(n) = { my(a, b) ; a=1; while(a^2+1Altug Alkan, Jan 13 2016

a(n) = A004214(n)^3. - Ray Chandler, Jan 29 2009

Definition corrected by Robert Israel, Jan 12 2016

A267321 Perfect powers that are not of the form x^2 + y^2 + z^2 where x, y and z are integers.

Original entry on oeis.org

343, 3375, 12167, 16807, 21952, 29791, 59319, 103823, 166375, 216000, 250047, 357911, 493039, 658503, 759375, 778688, 823543, 857375, 1092727, 1367631, 1404928, 1685159, 1906624, 2048383, 2460375, 2924207, 3442951, 3796416, 4019679, 4657463, 5359375, 6128487

Offset: 1

Altug Alkan, Jan 13 2016

nonn

Perfect powers that are the sum of 4 but no fewer nonzero squares. See first comment in A004215.
Intersection of A001597 and A004215.
A134738 is a subsequence.
Motivation for this sequence is the equation m^k = x^2 + y^2 + z^2 where x, y, z are integers and m > 0, k >= 2.
Corresponding exponents are 3, 3, 3, 5, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 3, 7, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 3, 3, 3, 3, 3, ...
Numbers of the form (4^i*(8*j+7))^(2*k+3) where i,j,k>=0. - Robert Israel, Jan 14 2016

			16807 is a term because 16807 = 7^5 and there is no integer values of x, y and z for the equation 7^5 = x^2 + y^2 + z^2.

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

Cf. A001597, A004215, A134738.

N:= 10^10; # to get all terms <= N
sort(convert({seq(seq(seq((4^i*(8*j+7))^(2*k+3),
    k=0..floor(1/2*(log[4^i*(8*j+7)](N)-3))),
     j = 0 .. floor((N^(1/3)*4^(-i)-7)/8)),
i=0..floor(log[4](N^(1/3)/7)))},list)); # Robert Israel, Jan 14 2016

isA004215(n) = { my(fouri, j) ; fouri=1 ; while( n >=7*fouri, if( n % fouri ==0, j= n/fouri -7 ; if( j % 8 ==0, return(1) ) ; ) ; fouri *= 4 ; ) ; return(0) ; } { for(n=1, 400, if(isA004215(n), print1(n, ", ") ; ) ; ) ; }
for(n=0, 1e7, if(isA004215(n) && ispower(n), print1(n, ", ")));

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A004215 Numbers that are the sum of 4 but no fewer nonzero squares.

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A134739 Cubes of (positive numbers that are not the sum of three nonzero squares), that is, the terms of A004214, cubed.

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A267321 Perfect powers that are not of the form x^2 + y^2 + z^2 where x, y and z are integers.

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Maple

PARI