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A004215 Numbers that are the sum of 4 but no fewer nonzero squares.
(Formerly M4349)
36
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 (list; graph; refs; listen; history; text; internal format)
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

REFERENCES

David S. Betts, Letter to N. J. A. Sloane, Nov 05 1976.

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. Sierpinski, 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

R. T. Bumby, Sums Of Four Squares

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

S. 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) = A044075(n)/2. Ray Chandler, Jan 30 2009

Numbers of the form 4^i(8j+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) = 6n + O(log n). - Charles R Greathouse IV, Dec 19 2013

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.

MATHEMATICA

Sort[Flatten[Table[4^i(8j + 7), {i, 0, 2}, {j, 0, 42}]]] (* Alonso del Arte, Jul 05 2005 *)

b = Table[x^3, {x, 1, 300}]; a = {}; Do[Do[Do[AppendTo[a, (x^2 + y^2 + z^2)^3], {x, 0, 30}], {y, 0, 30}], {z, 0, 30}]; Union[a]; k = Complement[b, a]; k^(1/3) (* Artur Jasinski, Nov 07 2007 *)

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

CROSSREFS

Cf. Complement A000378, A002828, A055039, A072401, A125084, A134738, A134739, A055045, A055046, A234000.

Sequence in context: A128840 A041935 A041092 * A206906 A179890 A043449

Adjacent sequences:  A004212 A004213 A004214 * A004216 A004217 A004218

KEYWORD

nonn,nice,easy

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

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Last modified July 23 23:07 EDT 2014. Contains 244873 sequences.