login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A112329 Number of divisors of n if n odd, number of divisors of n/4 if n divisible by 4, otherwise 0. 6
1, 0, 2, 1, 2, 0, 2, 2, 3, 0, 2, 2, 2, 0, 4, 3, 2, 0, 2, 2, 4, 0, 2, 4, 3, 0, 4, 2, 2, 0, 2, 4, 4, 0, 4, 3, 2, 0, 4, 4, 2, 0, 2, 2, 6, 0, 2, 6, 3, 0, 4, 2, 2, 0, 4, 4, 4, 0, 2, 4, 2, 0, 6, 5, 4, 0, 2, 2, 4, 0, 2, 6, 2, 0, 6, 2, 4, 0, 2, 6, 5, 0, 2, 4, 4, 0, 4, 4, 2, 0, 4, 2, 4, 0, 4, 8, 2, 0, 6, 3, 2, 0, 2, 4, 8 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

First occurrence of k: 2, 1, 3, 9, 15, 64, 45, 256, 96, 144, 192, 4096, 240, ????, 768, 576, 480, ????, 720, ..., . See A246063. (* Robert G. Wilson v, Oct 31 2013 *)

a(n) is the number of pairs (u, v) in NxZ satisfying u^2-v^2=n. See Kühleitner. - Michel Marcus, Jul 30 2017

REFERENCES

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, p. 142.

LINKS

Ray Chandler, Table of n, a(n) for n = 1..10000

M. Kühleitner, An Omega Theorem on Differences of Two Squares, Acta Mathematica Universitatis Comenianae, Vol. 61, 1 (1992) pp. 117-123. See Lemma 1 p. 2.

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

FORMULA

Multiplicative with a(2^e) = e-1 if e>0, a(p^e) = 1+e if p>2.

G.f.: Sum_{k>0} x^k / (1 - (-x)^k) = Sum_{k>0} -(-x)^k / (1 + (-x)^k).

Möbius transform is period 4 sequence [ 1, -1, 1, 1, ...].

sum(k>=1, x^(k^2) * (1+x^(2*k))/(1-x^(2*k)) ). - Joerg Arndt, Nov 08 2010

a(4*n + 2) = 0. a(n) = -(-1)^n * A048272(n). a(2*n - 1) = A099774(n). a(4*n) = A000005(n). a(4*n + 1) = A000005(4*n + 1). a(4*n - 1) = 2 * A078703(n).

a(n) = A094572(n) / 2. - Ray Chandler, Aug 23 2014

Bisection: a(2*k-1) = A000005(2*k-1), a(2*k) = A183063(2*k) - A001227(2*k), k >= 1. See the Hardy reference, p. 142 where a(n) = sigma^*_0(n). - Wolfdieter Lang, Jan 07 2017

a(n) = d(n) - 2*d(n/2) + 2*d(n/4) where d(n) = 0 if n is not an integer. See Kühleitner.

EXAMPLE

x + 2*x^3 + x^4 + 2*x^5 + 2*x^7 + 2*x^8 + 3*x^9 + 2*x^11 + 2*x^12 + ...

MAPLE

f:= proc(n) if n::odd then numtheory:-tau(n) elif n mod 4 = 0 then numtheory:-tau(n/4) else 0 fi end proc;

seq(f(i), i=1..100); # Robert Israel, Aug 24 2014

MATHEMATICA

Rest[ CoefficientList[ Series[ Sum[x^k/(1 - (-x)^k), {k, 111}], {x, 0, 110}], x]] (* Robert G. Wilson v, Sep 20 2005 *)

Table[If[OddQ[n], DivisorSigma[0, n], If[OddQ[n/2], 0, DivisorSigma[0, n/4]]], {n, 100} ] (* Ray Chandler, Aug 23 2014 *)

PROG

(PARI) {a(n) = if( n<1, 0, (-1)^n * sumdiv( n, d, (-1)^d))}

(PARI) {a(n) = if( n<1, 0, if( n%2, numdiv(n), if( n%4, 0, numdiv(n/4))))} /* Michael Somos, Sep 02 2006 */

(PARI) d(n) = if (denominator(n)==1, numdiv(n), 0);

a(n) = numdiv(n) - 2*d(n/2) + 2*d(n/4); \\ Michel Marcus, Jul 30 2017

CROSSREFS

Cf. A000005, A001227, A048272, A078703, A094572, A099774, A183063.

Sequence in context: A029338 A240883 A048272 * A325033 A117448 A093321

Adjacent sequences:  A112326 A112327 A112328 * A112330 A112331 A112332

KEYWORD

nonn,mult

AUTHOR

Michael Somos, Sep 04 2005

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 23 20:23 EDT 2019. Contains 323528 sequences. (Running on oeis4.)