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A046897 Sum of divisors of n that are not divisible by 4. 27
1, 3, 4, 3, 6, 12, 8, 3, 13, 18, 12, 12, 14, 24, 24, 3, 18, 39, 20, 18, 32, 36, 24, 12, 31, 42, 40, 24, 30, 72, 32, 3, 48, 54, 48, 39, 38, 60, 56, 18, 42, 96, 44, 36, 78, 72, 48, 12, 57, 93, 72, 42, 54, 120, 72, 24, 80, 90, 60, 72, 62, 96, 104, 3, 84, 144, 68, 54, 96, 144, 72 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

The o.g.f. is (theta_3(0,x)^4 - 1)/8, see the Hardy reference, eqs. 9.2.1, 9.2.3 and 9.2.4 on p. 133 for Sum' m*u_m. Also Hardy-Wright, p. 314. See also the Somos, Jan 25 2008 formula below. - Wolfdieter Lang, Dec 11 2016

REFERENCES

J. M. Borwein, D. H. Bailey and R. Girgensohn, Experimentation in Mathematics, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See p. 194.

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island 2002, p. 133.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Clarendon Press, Oxford, Fifth edition, 1979, p. 314.

P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 31, Article 273.

C. J. Moreno and S. S. Wagstaff, Jr., Sums of Squares of Integers, Chapman & Hall, 2006

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

a(n) = (-1)^(n+1)*Sum_{d divides n} (-1)^(n/d+d)*d. Multiplicative with a(2^e) = 3, a(p^e) = (p^(e+1)-1)/(p-1) for an odd prime p. - Vladeta Jovovic, Sep 10 2002 [For a proof of the multiplicative property, see for example Moreno and Wagstaff, p. 33. - N. J. A. Sloane, Nov 09 2016]

G.f.: Sum_{k>0} x^k/(1+(-x)^k)^2, or Sum_{k>0} k*x^k/(1+(-x)^k). - Vladeta Jovovic, Dec 16 2002

Expansion of (1 - phi(q)^4) / 8 in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Jan 25 2008

Equals inverse Mobius transform (A051731) of "count, 4*n = 0": (1, 2, 3, 0, 5, 6, 7, 0,...). - Gary W. Adamson, Jul 03 2008

A000118(n) = 8*a(n) for all n>0.

Dirichlet g.f.: (1 - 4^(1-s)) * zeta(s) * zeta(s-1). - Michael Somos, Oct 21 2015

EXAMPLE

G.f. = q + 3*q^2 + 4*q^3 + 3*q^4 + 6*q^5 + 12*q^6 + 8*q^7 + 3*q^8 + 13*q^9 + ...

MAPLE

A046897 := proc(n) if n mod 4 = 0 then numtheory[sigma](n)-4*numtheory[sigma](n/4) ; else numtheory[sigma](n) ; end if; end proc: # R. J. Mathar, Mar 23 2011

MATHEMATICA

a[n_] := Sum[ Boole[ !Divisible[d, 4]]*d, {d, Divisors[n]}]; Table[ a[n], {n, 1, 71}] (* Jean-Fran├žois Alcover, Dec 12 2011 *)

DivisorSum[#1, # &, Mod[#, 4] != 0 &] & /@ Range[71] (* Jayanta Basu, Jun 30 2013 *)

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q]^4 - 1) / 8, {q, 0, n}]; (* Michael Somos, Dec 30 2014 *)

PROG

(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, if(d%4, d)))};

(MAGMA) A := Basis( ModularForms( Gamma0(4), 2), 72); B<q> := (A[1] - 1)/8 + A[2]; B; /* Michael Somos, Dec 30 2014 */

(Haskell)

a046897 1 = 1

a046897 n = product $ zipWith

            (\p e -> if p == 2 then 3 else div (p ^ (e + 1) - 1) (p - 1))

            (a027748_row n) (a124010_row n)

-- Reinhard Zumkeller, Aug 12 2015

CROSSREFS

Cf. A000118, A051731, A069733.

Cf. A027748, A124010.

Sequence in context: A073181 A183100 * A109506 A000113 A069915 A033634

Adjacent sequences:  A046894 A046895 A046896 * A046898 A046899 A046900

KEYWORD

nonn,mult

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified July 22 22:25 EDT 2017. Contains 289676 sequences.