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A007442
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Inverse binomial transform of primes.
(Formerly M0065)
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5
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2, 1, 1, -1, 3, -9, 23, -53, 115, -237, 457, -801, 1213, -1389, 445, 3667, -15081, 41335, -95059, 195769, -370803, 652463, -1063359, 1570205, -1961755, 1560269, 1401991, -11023119, 36000427, -93408425, 214275735, -450374071
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| a(n) is the (n-1)-st difference of the first n primes. Although the magnitude of the terms appears to grow exponentially, a plot shows that the sequence a(n)/2^n has quite a bit of structure. See A082594 for an interesting application. - T. D. Noe (noe(AT)sspectra.com), May 09 2003
Graph this divided by A122803 using plot2 ! - Franklin T. Adams-Watters
Contribution from Eric Desbiaux (moongerms(AT)wanadoo.fr), Mar 25 2010: (Start)
It appears that : The "1"'s line in the Gilbreath Conjecture (GC),
is transformed into Inverse binomial transform of primes,
if the rule of the absolute values is deleted.
In the GC without prime's rule, 4 can replace 2 (4,3,5,7,11,P...)
with no effect on 1's line, and in this case A007442 undergoes +2 or -2. (End)
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REFERENCES
| N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| Franklin T. Adams-Watters, Table of n, a(n) for n = 1..1000
T. D. Noe, Plot of A007442
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Binomial Transform
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FORMULA
| a(n) = Sum_{k=0..n-1} (-1)^(n-k-1) binomial(n-1, k) prime(k+1)
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EXAMPLE
| a(4) = 7 - 3*5 + 3*3 - 2 = -1.
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MATHEMATICA
| Diff[lst_List] := Table[lst[[i + 1]] - lst[[i]], {i, Length[lst] - 1}]; n=1000; dt = Prime[Range[n]]; a = Range[n]; a[[1]] = 2; Do[dt = Diff[dt]; a[[i]] = dt[[1]], {i, 2, n}]; a
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CROSSREFS
| Cf. A082594.
Sequence in context: A112707 A196017 A054252 * A054772 A085384 A067856
Adjacent sequences: A007439 A007440 A007441 * A007443 A007444 A007445
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KEYWORD
| sign,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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