|
| |
|
|
A004215
|
|
Numbers that are the sum of 4 but no fewer nonzero squares.
(Formerly M4349)
|
|
32
| |
|
|
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; internal format)
|
|
|
|
OFFSET
| 1,1
|
|
|
COMMENTS
| 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 (mymontain(AT)yahoo.com), Nov 23 2006
Numbers whose cubes do not have a partition as a sum of 3 squares. a(n)^3=A134738(n) - Artur Jasinski (grafix(AT)csl.pl), Nov 07 2007
|
|
|
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.
L. J. Mordell, A new Waring's problem with squares of linear forms, Quart. J. Math., 1 (1930), 276-288 (see p. 283).
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).
S. Uchiyama, A five-square theorem, Publ. Res. Math. Sci., Vol 13, Number 1 (1977), 301-305.
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
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 (mathar(AT)strw.leidenuniv.nl), Nov 29 2006
|
|
|
MATHEMATICA
| Sort[Flatten[Table[4^i(8j + 7), {i, 0, 2}, {j, 0, 42}]]] (* From Alonso Delarte, 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) (* From Artur Jasinski, Nov 07 2007 *)
Select[Range[120], Mod[ #/4^IntegerExponent[ #, 4], 8] == 7 &] (* From 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 (mathar(AT)strw.leidenuniv.nl), Nov 22 2006
(PARI) isA004215(n)={ n\4^valuation(n, 4)%8==7 } \\ - M. F. Hasler, Mar 18 2011
|
|
|
CROSSREFS
| Cf. A000378, A002828, A055039, A072401, A125084, A134738, A134739.
Sequence in context: A128840 A041935 A041092 * A179890 A043449 A136768
Adjacent sequences: A004212 A004213 A004214 * A004216 A004217 A004218
|
|
|
KEYWORD
| nonn,nice
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com) and J. H. Conway (conway(AT)math.princeton.edu)
|
|
|
EXTENSIONS
| More terms from Arlin Anderson (starship1(AT)gmail.com). Additional comments from Jud McCranie (JudMcCranie(AT)ugaalum.uga.edu), Mar 19 2000.
|
| |
|
|
