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A036262
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Triangle of numbers arising from Gilbreath's conjecture: successive absolute differences of primes.
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19
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2, 1, 3, 1, 2, 5, 1, 0, 2, 7, 1, 2, 2, 4, 11, 1, 2, 0, 2, 2, 13, 1, 2, 0, 0, 2, 4, 17, 1, 2, 0, 0, 0, 2, 2, 19, 1, 2, 0, 0, 0, 0, 2, 4, 23, 1, 2, 0, 0, 0, 0, 0, 2, 6, 29, 1, 0, 2, 2, 2, 2, 2, 2, 4, 2, 31, 1, 0, 0, 2, 0, 2, 0, 2, 0, 4, 6, 37, 1, 0, 0, 0, 2, 2, 0, 0, 2, 2, 2, 4, 41, 1, 0, 0, 0, 0, 2, 0, 0, 0
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| The conjecture is that the leading term is always 1.
Odlyzko has checked it for primes up to pi(10^13) = 3*10^11.
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REFERENCES
| R. K. Guy, Unsolved Problems Number Theory, A10.
R. B. Killgrove and K. E. Ralston, On a conjecture concerning the primes, Math.Tables Aids Comput. 13(1959), 121-122.
H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Amer. Math. Soc., 1996, p. 208.
F. Proth, Sur la serie des nombres premiers, Nouv. Corresp. Math., 4 (1878) 236-240.
W. Sierpinski, L'induction incomplete dans la theorie des nombres, Scripta Math. 28 (1967), 5-13.
C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 410.
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LINKS
| T. D. Noe, Rows n=1..100 of triangle, flattened
A. M. Odlyzko, Iterated absolute values of differences of consecutive primes, Math. Comp. 61 (1993), 373-380.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
Eric Weisstein's World of Mathematics, Gilbreath's Conjecture
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EXAMPLE
| Table begins (conjecture is leading term is always 1):
2 3 5 7 11 13 17 19 23 ...
1 2 2 4 2, 4, 2, 4 ...
1 0 2 2 2, 2, 2 ...
1 2 0 0 0, 0 ...
1 2 0 0 0 ...
1 2 0 0 ...
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MATHEMATICA
| max = 14; triangle = NestList[ Abs[ Differences[#]] &, Prime[ Range[max]], max]; Flatten[ Table[ triangle[[n - k + 1, k]], {n, 1, max}, {k, 1, n}]] (* From Jean-François Alcover, Nov 04 2011 *)
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PROG
| (Haskell)
a036262 n k = delta !! (n - k) !! (k - 1) where delta = iterate
(\pds -> zipWith (\x y -> abs (x - y)) (tail pds) pds) a000040_list
-- Reinhard Zumkeller, Jan 23 2011
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CROSSREFS
| Cf. A001223, A036261, A036277, A054977.
Sequence in context: A133737 A125047 A045898 * A169613 A176572 A168017
Adjacent sequences: A036259 A036260 A036261 * A036263 A036264 A036265
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KEYWORD
| tabl,easy,nice,nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 23 2003
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