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A036262
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Array of numbers read by upward antidiagonals, arising from Gilbreath's conjecture: leading row lists the primes; the following rows give absolute values of differences of previous row.
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42
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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
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OFFSET
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0,1
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COMMENTS
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The conjecture is that the leading term is always 1.
Odlyzko has checked it for primes up to pi(10^13) = 3*10^11.
The second column, omitting the initial 3, is given in A089582. The number of "0"s preceding the first term > 1 in the n-th row is given in A213014. The first term > 1 in any row must equal 2, else the conjecture is violated: Obviously all terms except for the first one are even. Thus, if the 2nd term in some row is > 2, it is >= 4, and the first term of the subsequent row is >= 3. If there is a positive number of zeros preceding a first term > 2 (thus >= 4), this "jump" will remain constant and "propagate" (in subsequent rows) to the beginning of the row, and the previously discussed case applies.
The previous statement can also be formulated as: Gilbreath's conjecture is equivalent to: A036277(n) > A213014(n)+2 for all n.
CAVEAT: While table A036261 starts with the first absolute differences of the primes in its first row, the present sequence has the primes themselves in its uppermost row, which is sometimes referred to as "row 0". Thus, "first row" of this table A036262 may either refer to row 1 (1,2,2,...), or to row 0 (2,3,5,7,...), while the latter might, however, as well be referred to "row 1 of A036262" in other sequences or papers.
(End)
Suppose that S = (s(k)), for k >= 1, is a sequence of real numbers. For n >= 1, let g(1,n) = |s(n+1)-s(n)| and g(k,n) = |g(k-1,n+1) - g(k-1,n)| for k >= 2.
Call (g(k,n)) the Gilbreath array of S. Call the first column of this array the Gilbreath transform of S. Denote this transform by G(S), so that G(S) is the sequence (g(n,1)). If S is the sequence of primes, then the Gilbreath conjecture holds that G(S) consists exclusively of 1's. More generally, it appears that there are many S such that G(S) is eventually periodic. See A358691 for conjectured examples. (End)
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REFERENCES
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R. K. Guy, Unsolved Problems Number Theory, A10.
H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Amer. Math. Soc., 1996, p. 208.
W. Sierpiński, L'induction incomplète dans la théorie 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
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N. J. A. Sloane, New Gilbreath Conjectures, Sum and Erase, Dissecting Polygons, and Other New Sequences, Doron Zeilberger's Exper. Math. Seminar, Rutgers, Sep 14 2023: Video, Slides, Updates. (Mentions this sequence.)
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FORMULA
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EXAMPLE
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The array begins (conjecture is leading term is always 1):
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101
1 2 2 4 2 4 2 4 6 2 6 4 2 4 6 6 2 6 4 2 6 4 6 8 4 2
1 0 2 2 2 2 2 2 4 4 2 2 2 2 0 4 4 2 2 4 2 2 2 4 2 2
1 2 0 0 0 0 0 2 0 2 0 0 0 2 4 0 2 0 2 2 0 0 2 2 0 0
1 2 0 0 0 0 2 2 2 2 0 0 2 2 4 2 2 2 0 2 0 2 0 2 0 0
1 2 0 0 0 2 0 0 0 2 0 2 0 2 2 0 0 2 2 2 2 2 2 2 0 8
1 2 0 0 2 2 0 0 2 2 2 2 2 0 2 0 2 0 0 0 0 0 0 2 8 8
1 2 0 2 0 2 0 2 0 0 0 0 2 2 2 2 2 0 0 0 0 0 2 6 0 8
1 2 2 2 2 2 2 2 0 0 0 2 0 0 0 0 2 0 0 0 0 2 4 6 8 6
1 0 0 0 0 0 0 2 0 0 2 2 0 0 0 2 2 0 0 0 2 2 2 2 2 4
...
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MAPLE
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option remember ;
if n = 0 then
ithprime(k) ;
else
abs(procname(n-1, k+1)-procname(n-1, k)) ;
end if;
end proc:
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MATHEMATICA
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max = 14; triangle = NestList[ Abs[ Differences[#]] &, Prime[ Range[max]], max]; Flatten[ Table[ triangle[[n - k + 1, k]], {n, 1, max}, {k, 1, n}]] (* Jean-François Alcover, Nov 04 2011 *)
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PROG
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(Haskell)
a036262 n k = delta !! (n - k) !! (k - 1) where delta = iterate
(\pds -> zipWith (\x y -> abs (x - y)) (tail pds) pds) a000040_list
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CROSSREFS
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See A255483 for an interesting generalization.
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KEYWORD
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AUTHOR
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EXTENSIONS
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More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 23 2003
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STATUS
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approved
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