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A004215
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Numbers that are the sum of 4 but no fewer nonzero squares.
(Formerly M4349)
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93
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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
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OFFSET
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1,1
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COMMENTS
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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
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
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REFERENCES
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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.
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LINKS
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FORMULA
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Numbers of the form 4^i*(8*j+7), i >= 0, j >= 0. [A.-M. Legendre & C. F. Gauss]
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EXAMPLE
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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.
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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..]
(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
(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)))
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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EXTENSIONS
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More terms from Arlin Anderson (starship1(AT)gmail.com)
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STATUS
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approved
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