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
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
Named after the Austrian mathematician Johann Radon (1887-1956) and the German mathematician Adolf Hurwitz (1859-1919). - Amiram Eldar, Jun 15 2021
REFERENCES
T. Y. Lam, The Algebraic Theory of Quadratic Forms. Benjamin, Reading, MA, 1973, p. 131.
Takashi Ono, Variations on a Theme of Euler, Plenum, NY, 1994, p. 192.
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, Vol. 1 (1962), pp. 63-65.
J. Frank Adams, Vector fields on spheres, Bull. Amer. Math. Soc., Vol. 68 (1962), pp. 39-41.
J. Frank Adams, Vector fields on spheres, Annals of Math., Vol. 75 (1962), pp. 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., Vol. 307 (2003), pp. 3-29.
Adolf Hurwitz, Uber die Komposition der quadratischen formen, Math. Annalen, Vol. 88 (1923), pp. 1-25.
Michel A. Kervaire, Non-parallelizability of the sphere for n > 7, Proc. Nat. Acad. Sci. USA, Vol. 44, No. 3 (1958), pp. 280-283.
John Milnor, Some consequences of a theorem of Bott, Annals of Mathematics, Second Series, Vol. 68, No. 2 (1958), pp. 444-449.
Johann Radon, Lineare scharen orthogonaler matrizen,Abh. Math. Sem. Univ. Hamburg, Vol. 1 (1922), pp. 1-14.
Daniel B. Shapiro, Letter to N. J. A. Sloane, 1974.
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(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
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
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 8/3. - Amiram Eldar, Oct 22 2022
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-François 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
(Python)
def A003484(n): return (((m:=(~n&n-1).bit_length())&-4)<<1)+(1<<(m&3)) # Chai Wah Wu, Jul 09 2022
CROSSREFS
KEYWORD
nonn,easy,core,nice,mult
AUTHOR
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), Mar 20 2000
STATUS
approved