A003484 - OEIS

%I M0161 #103 Oct 22 2022 08:05:41

%S 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,

%T 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,

%U 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

%N Radon function, also called Hurwitz-Radon numbers.

%C 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

%C 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

%C Named after the Austrian mathematician Johann Radon (1887-1956) and the German mathematician Adolf Hurwitz (1859-1919). - _Amiram Eldar_, Jun 15 2021

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

%D Takashi Ono, Variations on a Theme of Euler, Plenum, NY, 1994, p. 192.

%D A. R. Rajwade, Squares, Camb. Univ. Press, London Math. Soc. Lecture Notes Series 171, 1993; see p. 127.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A003484/b003484.txt">Table of n, a(n) for n = 1..10000</a>

%H J. Frank Adams, <a href="http://dx.doi.org/10.1016/0040-9383(62)90096-4">Vector fields on spheres</a>, Topology, Vol. 1 (1962), pp. 63-65.

%H J. Frank Adams, <a href="https://doi.org/10.1090/S0002-9904-1962-10693-4">Vector fields on spheres</a>, Bull. Amer. Math. Soc., Vol. 68 (1962), pp. 39-41.

%H J. Frank Adams, <a href="http://www.jstor.org/stable/1970213">Vector fields on spheres</a>, Annals of Math., Vol. 75 (1962), pp. 603-632.

%H J.-P. Allouche and J. Shallit, <a href="http://www.math.jussieu.fr/~allouche/kreg2.ps">The Ring of k-regular Sequences, II</a>.

%H J.-P. Allouche and J. Shallit, <a href="http://dx.doi.org/10.1016/S0304-3975(03)00090-2">The ring of k-regular sequences, II</a>, Theoret. Computer Sci., Vol. 307 (2003), pp. 3-29.

%H Adolf Hurwitz, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002269074">Uber die Komposition der quadratischen formen</a>, Math. Annalen, Vol. 88 (1923), pp. 1-25.

%H Michel A. Kervaire, <a href="http://www.pnas.org/content/44/3/280.full.pdf">Non-parallelizability of the sphere for n > 7</a>, Proc. Nat. Acad. Sci. USA, Vol. 44, No. 3 (1958), pp. 280-283.

%H John Milnor, <a href="http://www.jstor.org/stable/1970255">Some consequences of a theorem of Bott</a>, Annals of Mathematics, Second Series, Vol. 68, No. 2 (1958), pp. 444-449.

%H Johann Radon, <a href="https://doi.org/10.1007/BF02940576">Lineare scharen orthogonaler matrizen</a>,Abh. Math. Sem. Univ. Hamburg, Vol. 1 (1922), pp. 1-14.

%H Daniel B. Shapiro, <a href="/A003484/a003484.pdf">Letter to N. J. A. Sloane, 1974</a>.

%H <a href="/index/Cor#core">Index entries for "core" sequences</a>.

%F a(n) = A003485(A007814(n)).

%F If n=2^(4*b+c)*d, 0<=c<=3, d odd, then a(n) = 8*b + 2^c.

%F 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).

%F 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

%F 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

%F 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

%F a((2*n-1)*2^p) = A003485(p), p >=0. - _Johannes W. Meijer_, Jun 07 2011, Dec 15 2012

%F 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

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 8/3. - _Amiram Eldar_, Oct 22 2022

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

%p 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

%p 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

%t a[n_] := 8*Quotient[IntegerExponent[n, 2], 4] + 2^Mod[IntegerExponent[n, 2], 4]; Table[a[n], {n, 1, 102}] (* _Jean-François Alcover_, Sep 08 2011, after _Paul D. Hanna_ *)

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

%o (Haskell)

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

%o -- _Reinhard Zumkeller_, Mar 11 2012

%o (Python)

%o def A003484(n): return (((m:=(~n&n-1).bit_length())&-4)<<1)+(1<<(m&3)) # _Chai Wah Wu_, Jul 09 2022

%Y See A053381 for a closely related sequence.

%Y Cf. A003485, A006519, A007814, A101119.

%K nonn,easy,core,nice,mult

%O 1,2

%A _N. J. A. Sloane_

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