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A036262 Triangle of numbers arising from Gilbreath's conjecture: successive absolute differences of primes read by anti-diagonals upwards. 22
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; text; internal format)
OFFSET

0,1

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.

From M. F. Hasler, Jun 02 2012: (Start)

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)

REFERENCES

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. Sierpinski, 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.

LINKS

T. D. Noe, Table of n, a(n) for n = 0..5049

R. B. Killgrove and K. E. Ralston, On a conjecture concerning the primes, Math.Tables Aids Comput. 13(1959), 121-122.

A. M. Odlyzko, Iterated absolute values of differences of consecutive primes, Math. Comp. 61 (1993), 373-380.

F. Proth, Sur la série des nombres premiers, Nouv. Corresp. Math., 4 (1878) 236-240.

W. Sierpinski, L'induction incomplète dans la théorie des nombres, Bulletin de la Société des mathématiciens et physiciens de la R.P de Serbie, Vol XIII, 1-2 (1961), Beograd, Yougoslavie.

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

FORMULA

T(0,k) = A000040(k). T(n,k) = |T(n-1,k+1)-T(n-1,k)|, n>0. - R. J. Mathar, Sep 19 2013

EXAMPLE

Table 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

...

MATHEMATICA

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 *)

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

CROSSREFS

Cf. A001223, A036261, A036277, A054977, A222310.

Sequence in context: A213371 A125047 A045898 * A169613 A176572 A168017

Adjacent sequences:  A036259 A036260 A036261 * A036263 A036264 A036265

KEYWORD

tabl,easy,nice,nonn

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 23 2003

STATUS

approved

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Last modified October 31 07:40 EDT 2014. Contains 248845 sequences.