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A003484
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Radon function, also called Hurwitz-Radon numbers.
(Formerly M0161)
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9
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1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 9, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 10, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 9, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 12, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 9, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 10, 1, 2, 1, 4, 1, 2
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Simon Plouffe (simon.plouffe(AT)gmail.com) observes that this sequence and A006519 (greatest power of 2 dividing n) are very similar, the difference being all zeros except for every 16-th term (see A101119 for nonzero differences). Dec 02, 2004.
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REFERENCES
| J. Frank Adams, Vector fields on spheres, Topology, 1 (1962), 63-65.
J. Frank Adams, Vector fields on spheres, Bull. Amer. Math. Soc. 68 (1962) 39-41.
J. Frank Adams, Vector fields on spheres, Annals of Math. 75 (1962) 603-632.
J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.
A. Hurwitz, Uber die Komposition der quadratischen formen, Math. Annalen 88 (1923) 1-25.
M. Kervaire, Non-parallelizability of the sphere for n > 7, Proc. Nat. Acad. Sci. USA 44 (1958) 280-283.
T. Y. Lam, The Algebraic Theory of Quadratic Forms. Benjamin, Reading, MA, 1973, p. 131.
J. Milnor, Some consequences of a theorem of Bott, Annals Math. 68 (1958) 444-449.
T. Ono, Variations on a Theme of Euler, Plenum, NY, 1994, p. 192.
J. Radon, Lineare Scharen Orthogonaler Matrizen, Abh. Math. Sem. Univ. Hamburg 1 (1922) 1-14.
A. R. Rajwade, Squares, Camb. Univ. Press, London Math. Soc. Lecture Notes Series 171, 1993; see p. 127.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| T. D. Noe, Table of n, a(n) for n = 1..10000
J.-P. Allouche and J. Shallit, The Ring of k-regular Sequences, II
Index entries for "core" sequences
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FORMULA
| If n=2^(4*b+c)*d, 0<=c<=3, d odd, then a(n) = 8*b + 2^c.
If n=2^m*d, d odd, then a(n) = 2*m+1 if m=0 mod 4, a(n) = 2*m if m=1 or 2 mod 4, a(n) = 2*m+2 (otherwise, i.e., if m=3 mod 4).
Multiplicative with a(p^e) = 2e + a_(e mod 4) if p = 2; 1 if p > 2; where a = (1, 0, 0, 2). - David W. Wilson (davidwwilson(AT)comcast.net), Aug 01, 2001.
Dirichlet g.f. zeta(s) *(1-1/2^s)* {7*2^(-4*s) +1 +2^(3-3*s) +3*2^(1-5*s) +2^(1-s) +2^(2-6*s) +2^(2-2*s) }/ (1-2^(-4*s))^2 . - R. J. Mathar, Mar 04 2011
a((2*k+1)*2^t) = A003485(t). [Johannes W. Meijer, Jun 7 2011]
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MAPLE
| readlib(ifactors): for n from 1 to 150 do if n mod 2 = 1 then printf(`%d, `, 1) fi: if n mod 2 = 0 then m := ifactors(n)[2][1][2]: if m mod 4 = 0 then printf(`%d, `, 2*m+1) fi: if m mod 4 = 1 then printf(`%d, `, 2*m) fi: if m mod 4 = 2 then printf(`%d, `, 2*m) fi: if m mod 4 = 3 then printf(`%d, `, 2*m+2) fi: fi: od: # from James A. Sellers Dec 07 2000
nmax:=102; pmax:=ceil(log(nmax)/log(2)): A003485 := proc(n): A003485(n) := ceil((n+1)/4) + ceil(n/4) + 2*ceil((n-1)/4) + 4*ceil((n-2)/4) end: for p from 0 to pmax do for n from 1 to nmax do A003484((2*n-1)*2^p):= A003485(p): od: od: seq(A003484(n), n=1..nmax); [Johannes W. Meijer, Jun 7 2011]
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MATHEMATICA
| a[n_] := 8*Quotient[IntegerExponent[n, 2], 4] + 2^Mod[IntegerExponent[n, 2], 4]; Table[a[n], {n, 1, 102}] (* From Jean-François Alcover, Sep 08 2011, after P. D. Hanna *)
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PROG
| (PARI) a(n)=8*(valuation(n, 2)\4)+2^(valuation(n, 2)%4) /* Paul D. Hanna (pauldhanna(AT)juno.com), Dec 02 2004 */
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CROSSREFS
| See A053381 for a closely related sequence. Cf. A003485.
a(n) = A003485(A007814(n)).
Cf. A006519, A101119.
Sequence in context: A133186 A084236 A068057 * A118827 A118830 A055975
Adjacent sequences: A003481 A003482 A003483 * A003485 A003486 A003487
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KEYWORD
| nonn,easy,core,nice,mult
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Larry Reeves (larryr(AT)acm.org), Mar 20 2000
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