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A111938 a(n) = n times number of divisors of n of form 4m+1 - n times number of divisors of form 4m+3. 0
1, 2, 0, 4, 10, 0, 0, 8, 9, 20, 0, 0, 26, 0, 0, 16, 34, 18, 0, 40, 0, 0, 0, 0, 75, 52, 0, 0, 58, 0, 0, 32, 0, 68, 0, 36, 74, 0, 0, 80, 82, 0, 0, 0, 90, 0, 0, 0, 49, 150, 0, 104, 106, 0, 0, 0, 0, 116, 0, 0, 122, 0, 0, 64, 260, 0, 0, 136, 0, 0, 0, 72, 146, 148, 0, 0, 0, 0, 0, 160, 81, 164 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Table of n, a(n) for n=1..82.

FORMULA

Multiplicative with a(p^e) = p^e if p = 2; (e+1)p^e if p == 1 (mod 4); (1+(-1)^e)/2 p^e if p == 3 (mod 4).

G.f.: Sum_{k>0} k(x^k-x^(3k))/(1+x^(2k))^2 = Sum_{k>0} -(-1)^k(2k-1)x^(2k-1)/(1-x^(2k-1))^2.

G.f.: xd/dx(theta_3(x)^2)/4. - Michael Somos, Nov 07 2005

G.f.: (1/4)* Sum_{u,v} (u*u +v*v)* x^(u*u +v*v). - Michael Somos, Jun 14 2007

PROG

(PARI) a(n)=if(n<1, 0, n*sumdiv(n, d, (d%4==1)-(d%4==3)))

(PARI) {a(n)=local(r); if(n<1, 0, r=sqrtint(n); sum(x=-r, r, sum(y=-r, r, if(x^2+y^2==n, (x+y)^2) ))/4 )} /* Michael Somos, Sep 12 2005 */

(PARI) {a(n)=if(n<1, 0, n*polcoeff( sum(k=1, sqrtint(n), 2*x^k^2, 1+x*O(x^n))^2, n)/4 )} /* Michael Somos, Sep 12 2005 */

CROSSREFS

n*A002654(n)=a(n).

Sequence in context: A287314 A077119 A002938 * A224822 A246928 A167341

Adjacent sequences:  A111935 A111936 A111937 * A111939 A111940 A111941

KEYWORD

nonn,mult

AUTHOR

Michael Somos, Aug 21 2005

STATUS

approved

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Last modified August 20 03:48 EDT 2019. Contains 326139 sequences. (Running on oeis4.)