A004215 Numbers that are the sum of 4 but no fewer nonzero squares. (Formerly M4349)

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

1,1

Comments

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

References

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.

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.

Links

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

Formula

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

Example

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.

Maple

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

Mathematica

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 *)

Prog

(PARI) 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
(PARI) isA004215(n)= n\4^valuation(n, 4)%8==7 \\ M. F. Hasler, Mar 18 2011
(Haskell)
a004215 n = a004215_list !! (n-1)
a004215_list = filter ((== 4) . a002828) [1..]
-- Reinhard Zumkeller, Feb 26 2015
(Python)
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
(Python)
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

Crossrefs

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.

Sequence in context: A349233 A335512 A369375 * A206906 A295669 A179890

Adjacent sequences: A004212 A004213 A004214 * A004216 A004217 A004218

Keyword

nonn,nice,easy,changed

Author

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

Extensions

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

Additional comments from Jud McCranie, Mar 19 2000

Status

approved