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A360284
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Least integer nu such that the first zero of the Bessel j-function of index nu is at least nu + n.
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1
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0, 2, 7, 16, 29, 48, 73, 106, 148, 199, 260, 333, 417, 515, 627, 754, 897, 1057, 1234, 1431, 1647, 1884, 2142, 2423, 2727, 3056, 3410, 3791, 4198, 4634, 5099, 5594, 6120, 6678, 7268, 7893, 8552, 9247, 9979, 10748, 11555, 12402, 13290
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OFFSET
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2,2
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COMMENTS
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Tricomi proved that the first zero of j_nu occurs at nu + a*nu^(1/3) + b*nu^(-1/3) + O(1/nu). The PARI program below uses an estimate with a = 1.85575708087 and b = 1.
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REFERENCES
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Francesco Tricomi, Sulle funzioni di Bellel di ordine e argomento pressochè uguali, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur., 83:3-20 (1949).
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LINKS
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FORMULA
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Tricomi (cited in Elbert & Laforgia and McCann) proved that a(n) ~ kn^3. It seems that k is approximately 0.15647199543.
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PROG
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(PARI) esta(n)=my(a=1.85575708087); ((n+sqrt(n^2-4*a))/2/a)^3
a(n)=if(n==2, return(0)); my(k=esta(n)\1, t=besseljzero(k)-k); if(t<n, while(besseljzero(k++)-k<n, ); k, while(besseljzero(k--)-k>=n, ); k+1)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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