Search: a013631 -id:a013631
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A033167
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Positions of the incrementally largest terms in the continued fraction expansion of zeta(3), offset 1 variant.
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+10
1
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1, 2, 4, 29, 63, 572, 1556, 2013, 2530, 2760, 3019, 4159, 4741, 6820, 10565, 11666, 32859, 139893, 392130, 707970, 1049722, 2081165, 14990404, 36112276, 39552835, 42710787, 199618806
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OFFSET
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1,2
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COMMENTS
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Positions in this sequence correspond to the n-th term of A013631 at index n-1.
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LINKS
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A013687
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Continued fraction for zeta(11).
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+10
0
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1, 2023, 1, 1, 12, 1, 2, 2, 1, 102, 1, 44, 1, 2, 2, 1, 2, 3, 1, 5, 2, 1, 1, 2, 1, 13, 4, 14, 2, 5, 1, 5, 1, 6, 1, 2, 9, 1, 1, 1, 1, 7, 1, 2, 3, 1, 39, 3, 119, 12, 1, 1, 5, 1, 1, 151, 3, 4, 1, 2, 4, 98, 29, 6, 2, 1, 3, 9, 1, 1, 1, 5, 1, 2
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OFFSET
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0,2
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LINKS
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MATHEMATICA
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CROSSREFS
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KEYWORD
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nonn,cofr
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A013689
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Continued fraction for zeta(13).
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+10
0
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1, 8149, 13, 1, 2, 1, 6, 23, 3, 1, 7, 1, 1, 5, 1, 1, 4, 1, 1, 1, 4, 1, 1, 2, 2, 8, 1, 29, 32, 22, 2, 123, 1, 2, 1, 10, 1, 2, 2, 1, 4, 1, 13, 5, 8, 34, 2, 1, 7, 1, 2, 1, 3, 20, 8, 1, 4, 1, 5, 1, 59, 3, 7, 1, 1, 3, 2, 6, 1, 1, 2, 9, 1, 1
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OFFSET
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0,2
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LINKS
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MATHEMATICA
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CROSSREFS
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KEYWORD
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nonn,cofr
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A013690
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Continued fraction for zeta(14).
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+10
0
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1, 16327, 36, 19, 2, 1, 35, 1, 4, 7, 5, 1, 1, 1, 3, 1, 2, 3, 2, 1, 3, 3, 1, 1, 2, 1, 3, 1, 1, 7, 1, 4, 7, 4, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 4, 9, 2, 2, 1, 23, 6, 1, 2, 1, 2, 1, 1, 10, 1, 19, 7, 1, 1, 42, 1, 15, 1, 1, 4, 1, 2, 2, 1
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OFFSET
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0,2
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LINKS
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MATHEMATICA
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CROSSREFS
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KEYWORD
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nonn,cofr
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A013691
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Continued fraction for zeta(15).
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+10
0
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1, 32692, 3, 3, 1, 4, 1, 2, 3, 2, 1, 1, 1, 1, 1, 3, 1, 5, 1, 4, 1, 54, 1, 5, 5, 1, 20, 57, 5, 8, 1, 2, 26, 1, 1, 1, 1, 10, 1, 12, 1, 1, 7, 1, 2, 4, 1, 4, 1, 3, 5, 1, 1, 1, 1, 2, 4, 1, 18, 2, 2, 4, 1, 7, 4, 5, 1, 4, 2, 1, 1, 3, 1, 5, 1, 28
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OFFSET
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0,2
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LINKS
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MATHEMATICA
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CROSSREFS
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KEYWORD
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nonn,cofr
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A013692
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Continued fraction for zeta(16).
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+10
0
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1, 65435, 2, 1, 5, 1, 4, 1, 3, 3, 1, 7, 1, 2, 6, 2, 1, 7, 1, 1, 2, 1, 4, 4, 2, 3, 13, 1, 2, 1, 5, 1, 1, 8, 1, 5, 1, 1, 1, 4, 1, 2, 2, 2, 1, 44, 1, 2, 1, 2, 4, 2, 1, 6, 153, 41, 1, 26, 1, 4, 1, 3, 3, 1, 1, 1, 5, 6, 15, 4, 7, 1, 1, 1, 2, 1
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OFFSET
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0,2
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LINKS
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MATHEMATICA
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CROSSREFS
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KEYWORD
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nonn,cofr
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A013693
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Continued fraction for zeta(17).
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+10
0
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1, 130938, 12, 2, 2, 8, 1, 6, 2, 3, 4, 2, 6, 1, 1, 7, 3, 10, 1, 5, 1, 2, 1, 2, 33, 3, 1, 4, 1, 1, 7, 5, 7, 1, 4, 1, 6, 1, 1, 2, 1, 1, 1, 5, 1, 1, 4, 1, 1, 1, 3, 1, 1, 3, 8, 2, 2, 2, 5, 5, 4, 2, 7, 2, 45, 5, 6, 2, 1, 1, 53, 1, 1, 1, 4, 1, 2
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listen;
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OFFSET
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0,2
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LINKS
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MATHEMATICA
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CROSSREFS
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KEYWORD
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nonn,cofr
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A265824
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Continued fraction expansion of the prime zeta function at 3.
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+10
0
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0, 5, 1, 2, 1, 1, 2, 17, 4, 1, 7, 1, 1, 5, 24, 1, 1, 2, 3, 11, 1, 3, 23, 1, 1, 2, 1, 3, 1, 6, 1, 4, 3, 3, 1, 2, 1, 4, 1, 1, 3, 1, 1, 1, 2, 23, 2, 6, 2, 2, 1, 1, 7, 3, 13, 1, 1, 2, 6, 1, 5, 5, 1, 2, 1, 1, 1, 1, 2, 1, 3, 1, 28, 2, 1, 4, 10, 3, 2, 1, 1, 2, 1, 3, 1, 1
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OFFSET
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0,2
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COMMENTS
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Continued fraction of Sum_{n>=1} 1/prime(n)^3 = 0.1747626392994435364231...
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LINKS
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EXAMPLE
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1/2^3 + 1/3^3 + 1/5^3 +1/7^3 + 1/11^3 + 1/13^3 +... = 1/(5 + 1/(1 + 1/(2 + 1/(1 + 1/(1 + 1/(2 + 1/(17 + 1/(4 + 1/…)))))))).
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MATHEMATICA
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ContinuedFraction[PrimeZetaP[3], 85]
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CROSSREFS
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KEYWORD
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nonn,cofr
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AUTHOR
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STATUS
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approved
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A269444
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Continued fraction expansion of the Dirichlet eta function at 3.
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+10
0
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0, 1, 9, 6, 2, 1, 1, 1, 1, 1, 1, 6, 1, 4, 1, 7, 2, 1, 1, 1, 2, 91, 32, 1, 1, 6, 23, 1, 1, 1, 1, 2, 9, 1, 2, 1, 1, 5, 1, 1, 37, 12, 1, 12, 3, 2, 87, 1, 4, 2, 2, 2, 320, 1, 7, 1, 2, 6, 3, 1, 6, 4, 1, 4, 2, 1, 69, 1, 4, 3, 3, 1, 14, 3, 1, 3, 1, 10, 2, 694, 2, 4, 21, 1, 1, 1, 3, 3, 10, 2, 1, 2, 2, 1, 3, 5, 1, 3, 9, 1
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OFFSET
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0,3
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COMMENTS
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Continued fraction expansion of Sum_{k>=1} (-1)^(k - 1)/k^3 = (3*zeta(3))/4 = 0.901542677369695714...
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LINKS
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EXAMPLE
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1/1^3 - 1/2^3 + 1/3^3 - 1/4^3 + 1/5^3 - 1/6^3 +... = 1/(1 + 1/(9 + 1/(6 + 1/(2 + 1/(1 + 1/(1 + 1/...)))))).
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MATHEMATICA
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ContinuedFraction[(3 Zeta[3])/4, 100]
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CROSSREFS
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KEYWORD
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nonn,cofr
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AUTHOR
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STATUS
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approved
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A343244
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Position of the first occurrence of an element in the continued fraction of zeta(n) which is larger than the second element.
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+10
0
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5, 4, 8, 14, 10, 63, 120, 79, 1270, 779, 1749, 3410, 13668, 17704, 20909, 175782, 127426
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OFFSET
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2,1
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COMMENTS
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a(20) = 111604.
The corresponding values of the a(n)-th elements are 4, 18, 183, 32, 61, 9283, 462, 1483, 3530, 3484, 10812, 8954, ...
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LINKS
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EXAMPLE
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The continued fraction of zeta(3) is [1; 4, 1, 18, 1, 1, ...]. The first element which is larger than 4 is 18 whose position is 4. Therefore, a(3) = 4.
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MATHEMATICA
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a[n_] := Module[{c = ContinuedFraction[Zeta[n], 10000]}, FirstPosition[c, _?(# > c[[2]] &)][[1]]]; Array[a, 10, 2]
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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