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Search: id:A119712
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| A119712 |
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a(k) is the smallest integer n such that the k-th difference of the partition sequence A000041 is positive from n onwards. |
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+0
1
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| 0, 1, 6, 23, 64, 129, 222, 345, 502, 695, 924, 1193, 1502, 1853, 2246, 2687, 3172, 3705, 4286, 4917, 5600, 6333, 7118, 7957, 8848, 9797, 10800, 11861, 12978, 14153, 15386, 16681, 18034, 19447, 20922, 22459, 24060, 25723, 27448, 29239, 31094, 33015
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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The first entry in considered to be indexed by zero. For example, the third difference A072380 starts with -1,1 and continues alternating in sign till the 24th entry, from which point it is positive.
Using a different definition of the difference operator, this sequence has also been given as 1,8,26,68,134,228,352, etc
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REFERENCES
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I. J. Good, Problem 6137, American Mathematical Monthly 1978 pages 830-831
Hansraj Gupta, Finite Differences of the Partition Function, pp. 1241-1243.
A. M. Odlyzko, Differences of the partition function, Acta Arith., 49 (1988), pp. 237-254
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FORMULA
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Odlyzko gives an asymptotic formula a(k)~(6/(Pi)^2) * (klogk)^2
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MAPLE
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with (combinat): DD:= proc(p) proc(n) option remember; p(n+1) -p(n) end end: a:= proc(n) option remember; local f, k; if n=0 then 0 else f:= (DD@@n)(numbpart); for k from a(n-1) while not (f(k)>0 and f(k+1)>0) do od; k fi end: seq (a(n), n=0..20); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Jul 20 2009]
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CROSSREFS
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Sequence in context: A026817 A162267 A009017 this_sequence A005745 A045618 A038737
Adjacent sequences: A119709 A119710 A119711 this_sequence A119713 A119714 A119715
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KEYWORD
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nonn
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AUTHOR
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Moshe Newman (moshnoiman(AT)gmail.com), Jun 11 2006
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EXTENSIONS
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a(11) - a(41) from Alois P. Heinz (heinz(AT)hs-heilbronn.de), Jul 20 2009
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