A003484 - OEIS

A003484

Radon function, also called Hurwitz-Radon numbers.
(Formerly M0161)

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

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.

Index entries for "core" sequences.

FORMULA

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

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

See A053381 for a closely related sequence.

Cf. A003485, A006519, A007814, A101119.

Sequence in context: A376645 A335852 A353751 * A118827 A118830 A055975

Adjacent sequences: A003481 A003482 A003483 * A003485 A003486 A003487