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Search: id:A111938
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| A111938 |
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a(n) = n times number of divisors of n of form 4m+1 - n times number of divisors of form 4m+3. |
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+0
1
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| 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; listen)
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OFFSET
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1,2
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FORMULA
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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
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PROGRAM
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(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 */
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CROSSREFS
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n*A002654(n)=a(n).
Sequence in context: A021492 A077119 A002938 this_sequence A167341 A055978 A069025
Adjacent sequences: A111935 A111936 A111937 this_sequence A111939 A111940 A111941
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KEYWORD
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nonn,mult
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AUTHOR
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Michael Somos, Aug 21 2005
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