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A003484 Radon function, also called Hurwitz-Radon numbers.
(Formerly M0161)
14
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; text; internal format)
OFFSET

1,2

COMMENTS

This sequence and A006519 (greatest power of 2 dividing n) are very similar, the difference being all zeros except for every 16th term (see A101119 for nonzero differences). - Simon Plouffe, Dec 02 2004

a(A005408(n))=1; a(2*n) = A209675(n); a(A016825(n))=2; a(A017113(n))=4; a(A051062(n))=8. - Reinhard Zumkeller, Mar 11 2012

For all n congruent to 2^k (mod 2^(k+1)), a(n) is the same. Therefore, for any natural number m, the list of the first 2^m - 1 terms is palindromic. - Ivan N. Ianakiev, Jul 21 2019

REFERENCES

T. Y. Lam, The Algebraic Theory of Quadratic Forms. Benjamin, Reading, MA, 1973, p. 131.

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).

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

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

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.

J. Milnor, Some consequences of a theorem of Bott, Annals Math. 68 (1958) 444-449.

D. B. Shapiro, Letter to N. J. A. Sloane, 1974

Index entries for "core" sequences

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, 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*n-1)*2^p) = A003485(p), p >=0. - Johannes W. Meijer, Jun 07 2011, Dec 15 2012

Lambert series g.f. Sum_(k >=0) q^(2^(4*k))/(1-q^(2^(4*k))) +q^(2^(4*k+1))/(1-q^(2^(4*k+1))) +2*q^(2^(4*k+2))/(1-q^(2^(4*k+2))) +4*q^(2^(4*k+3))/(1-q^(2^(4*k+3))). - Mamuka Jibladze, Dec 07 2016

EXAMPLE

G.f. = x + 2*x^2 + x^3 + 4*x^4 + x^5 + 2*x^6 + x^7 + 8*x^8 + x^9 + ...

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: # James A. Sellers, Dec 07 2000

nmax:=102; A003485 := proc(n): A003485(n) := ceil((n+1)/4) + ceil(n/4) + 2*ceil((n-1)/4) + 4*ceil((n-2)/4) end: A029837 := n -> ceil(simplify(log[2](n))): for p from 0 to A029837(nmax) do for n from 1 to ceil(nmax/(p+2)) do A003484((2*n-1)*2^p):= A003485(p): od: od: seq(A003484(n), n=1..nmax); # Johannes W. Meijer, Jun 07 2011, Dec 15 2012

MATHEMATICA

a[n_] := 8*Quotient[IntegerExponent[n, 2], 4] + 2^Mod[IntegerExponent[n, 2], 4]; Table[a[n], {n, 1, 102}] (* Jean-Fran├žois Alcover, Sep 08 2011, after Paul D. Hanna *)

PROG

(PARI) a(n)=8*(valuation(n, 2)\4)+2^(valuation(n, 2)%4) /* Paul D. Hanna, Dec 02 2004 */

(Haskell)

a003484 n = 2 * e + cycle [1, 0, 0, 2] !! e  where e = a007814 n

-- Reinhard Zumkeller, Mar 11 2012

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

KEYWORD

nonn,easy,core,nice,mult

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Larry Reeves (larryr(AT)acm.org), Mar 20 2000

STATUS

approved

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Last modified October 21 14:56 EDT 2019. Contains 328301 sequences. (Running on oeis4.)