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Search: id:A164655
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| A164655 |
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Numerators of partial sums of Theta(3):=sum(1/(2*j-1)^3,j=1..infty). |
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+0
2
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| 1, 28, 3527, 1213136, 32797547, 43684790932, 96017087247229, 96044168328256, 471956397645187853, 3237597973008257555852, 462561506842656976961, 5628425850334528955928112, 703596058798919360293439483, 18998011529681231695738912916, 463360571051954739540899597748949
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OFFSET
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1,2
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COMMENT
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The denominators look like those given for the partial sums of another series in A128507.
Rationals (partial sums) Theta(3,n) := sum(1/(2*j-1)^3,j=1..n) (in lowest terms). The limit of these rationals is Theta(3)= (1-1/2^3)*Zeta(3) approximately 1.051799790 (Zeta(n) is the Euler, Riemann Zeta function).
This is a member of the k-family of rational sequences Theta(k,n):=sum(1/(2*j-1)^k,j=1..n), k>=1, which coincides for k=1 with A025550/A025547 (but only for the first 38 entries), for k=2 with A120268/A128492, for k=3 with a(n)/A128507(n) (the denominators may depart for higher n values), A120269/A128493 and A164656/A164657, for k=4 and 5, respectively.
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REFERENCES
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R. Ayoub, Euler and the Zeta Function, Am. Math. Monthly 81 (1974) 1067-1086, p. 1070.
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LINKS
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W. Lang: Theta(k,n), k-family of rational sequences and limits.
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FORMULA
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a(n) = numer(Theta(3,n))= numerator(sum(1/(2*j-1)^3,j=1..n)), n>=1.
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EXAMPLE
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Rationals Theta(3,n): [1, 28/27, 3527/3375, 1213136/1157625, 32797547/31255875, 43684790932/41601569625,...].
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CROSSREFS
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Sequence in context: A107444 A061787 A128506 this_sequence A036525 A131315 A159407
Adjacent sequences: A164652 A164653 A164654 this_sequence A164656 A164657 A164658
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KEYWORD
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nonn,frac,easy
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de) Oct 16 2009
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