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A030199
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G.f.: x * Product( (1 - x^k)*(1 - x^(23*k)), k=1..infinity).
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2
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0, 1, -1, -1, 0, 0, 1, 0, 1, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, -1, 1, 1, 1, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, -1, -1, 1, 1, -1, 0, 0, 0, -1, 0, 0, 0, 1, 2, 0, 0, 1, 0, 1, 0, 0, 0, 0, -1, 0, -1, 0, -1, 0, -1, 0, 0, -1, 0, 0, -1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, -1, 0, 0, 2, 0, 0, -1
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,60
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REFERENCES
| M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
Y. Martin and K. Ono, Eta-quotients and elliptic curves, Proc. Amer. Math. Soc. 125 (1997), no. 11, 3169-3176.
J.-P. Serre, On a theorem of Jordan, Bull. Amer. Math. Soc., 40 (No. 4, 2003), 429-440, see p. 434.
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LINKS
| A. Granville and G. Martin, Prime number races
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FORMULA
| Expansion of eta(q) * eta(q^23) in powers of q.
Euler transform of period 23 sequence [ -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -2, ...]. - Michael Somos, May 02 2005
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = -v^3 + 2 *u * v * w + 2 * u * w^2 + u^2 * w. - Michael Somos, May 02 2005
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1^2 * u3 * u6 +2 * u1 * u2 * u3 * u6 - 2 * u1 * u6^3 + 2 * u2^2 * u3 * u6 - u2 * u3^3. - Michael Somos, May 02 2005
a(n) is multiplicative with a(23^e) = 1. Let y = number of zeros of x^3 - x - 1 modulo p, then a(p^e) = (1 + (-1)^e)/2 if y = 1, a(p^e) = e+1 if y = 3, a(p^e) = (e-1)%3 - 1 if y = 0. - Michael Somos Oct 19 2005
a(8n + 4) = a(23n + 5) = a(23n + 7) = a(23n + 10) = a(23n + 11) = a(23n + 14) = a(23n + 15) = a(23n + 19) = a(23n + 20) = a(23n + 21) = a(23n + 22) = 0. - Michael Somos Oct 19 2005
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EXAMPLE
| q - q^2 - q^3 + q^6 + q^8 -q ^13 - q^16 + q^23 - q^24 + q^25 + q^26 + ...
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PROG
| (PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^23 + A), n))} /* Michael Somos May 02 2005 */
(PARI) {a(n) = local(A, p, e, y); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], if( p = A[k, 1], e = A[k, 2]; if( p==23, 1, y = sum( x=1, p-1, (x^3 - x - 1)%p == 0); if( y==1, 1-e%2, if(y, e+1, (e-1)%3-1))))))} /* Michael Somos Oct 19 2005 */
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CROSSREFS
| Sequence in context: A116376 A165766 A102082 * A005089 A119395 A087476
Adjacent sequences: A030196 A030197 A030198 * A030200 A030201 A030202
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KEYWORD
| sign,mult
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Added the reference to Martin and Ono. - Chandan Singh Dalawat (dalawat(AT)gmail.com), Jul 23 2010
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