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A091400
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Product_{ p | N } (1 + Legendre(-1,p) ).
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4
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1, 1, 0, 1, 2, 0, 0, 1, 0, 2, 0, 0, 2, 0, 0, 1, 2, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 0, 2, 0, 0, 1, 0, 2, 0, 0, 2, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 1, 4, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 2, 0, 0, 2, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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COMMENTS
| Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A10054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
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REFERENCES
| G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971, see p. 25, Eq. (2) (but without the restriction that a(4k) = 0).
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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| Here we use the definition that Legendre(-1, 2) = 0, Legendre(-1, p) = 1 if p == 1 mod 4, = -1 if p == 3 mod 4. This is Shimura's definition, which is different from Maple's.
Moebius transform is period 36 sequence [ 1, 0, -1, 0, 1, 0, -1, 0, 0, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, 0, 0, 1, 0, -1, 0, 1, 0, -1, 0, ...]. - Michael Somos Apr 19 2007
Expansion of (phi(q)^2 -phi(q^9)^2)/4 in powers of q where phi() is a Ramanujan theta function.
a(n) is multiplicative with a(2^e) = 1, a(3^e) = 0^e, a(p^e) = (1+(-1)^e)/2 if p == 3 (mod 4), a(p^e) = e+1 if p == 1 (mod 4).
a(2n)= a(n). a(3n)= a(4n+3)= 0.
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MAPLE
| with(numtheory): A091400 := proc(n) local i, t1, t2; t1 := ifactors(n)[2]; t2 := 1; for i from 1 to nops(t1) do if t1[i][1] > 2 then t2 := t2*(1+legendre(-1, t1[i][1])); fi; od: t2; end;
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PROG
| (PARI) {a(n)= if(n<1, 0, abs( sumdiv(n, d, kronecker(12, d)* kronecker(-3, n/d)) ))} /* Michael Somos Apr 19 2007 */
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CROSSREFS
| Cf. A091379.
A122865(n)= a(3n+1). A122856(n)= a(3n+2). |A129448(n)|= a(n).
Sequence in context: A092303 A063725 A084888 * A129448 A123759 A072453
Adjacent sequences: A091397 A091398 A091399 * A091401 A091402 A091403
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KEYWORD
| nonn,mult
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Mar 02 2004
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