A222310 -id:A222310 - cp's OEIS frontend

A222311 Leading diagonal of triangle in A222310.

1, 2, 3, 6, 5, 15, 105, 70, 1, 5, 33, 55, 65, 273, 1001, 1430, 17, 17, 969, 4845, 1785, 6545, 37145, 81719, 17, 1105, 3553, 969969, 672945, 81345, 955049953, 66786710, 33, 561, 385, 6545, 6105, 657305, 15873, 8544965, 1353, 268345, 61705, 329681, 650793, 24173705985, 3065857, 250538768183, 561, 33, 21945

Offset: 1

N. J. A. Sloane, Feb 16 2013

nonn

See A222313 for the numbers that appear in this sequence.

N. J. A. Sloane, Table of n, a(n) for n = 1..500
C. Cobeli and A. Zaharescu, Promenade around Pascal Triangle-Number Motives, Bull. Math. Soc. Sci. Math. Roumanie, Tome 56(104) No. 1, 2013, 73-98.
Cristian Cobeli, Alexandru Zaharescu, A game with divisors and absolute differences of exponents, arXiv:1411.1334 [math.NT], 2014 (see page 12).
Cristian Cobeli, Mihai Prunescu, Alexandru Zaharescu, A growth model based on the arithmetic Z-game, arXiv:1511.04315 [math.NT], 2015.

Cf. A222310, A222313.

Join[r = {1}, Table[Reverse[r = FoldList[#1*#2/GCD[#1, #2]^2&, n, r]], {n, 2, 100}][[All, 1]]] (* Jean-François Alcover, Sep 04 2018, after Ivan Neretin in A222310 *)

A036262 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.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

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)
From Clark Kimberling, Nov 27 2022: (Start)
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)

			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
...

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.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 192.
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.

T. D. Noe, Table of n, a(n) for n = 0..5049
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
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. Sierpiński, 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).
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.)
Eric Weisstein's World of Mathematics, Gilbreath's Conjecture
Index entries for sequences related to Gilbreath conjecture and transform

Cf. A001223, A036261, A036277, A054977, A222310, A358691, A089582 (2nd col).

See A255483 for an interesting generalization.

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

A036262 := proc(n, k)
    option remember ;
    if n = 0 then
        ithprime(k) ;
    else
        abs(procname(n-1, k+1)-procname(n-1, k)) ;
    end if;
end proc:
seq(seq( A036262(d-k,k),k=1..d),d=1..13) ; # R. J. Mathar, May 10 2023

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 *)

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

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

Definition edited by N. J. A. Sloane, May 03 2023

A222313 A222311 sorted and duplicates removed (conjectured).

Original entry on oeis.org

1, 2, 3, 5, 6, 15, 17, 33, 41, 55, 57, 65, 70, 105, 129, 257, 273, 385, 561, 897, 969, 1001, 1105, 1353, 1430, 1785, 2049, 2145, 2337, 2665, 3553, 4097, 4305, 4745, 4845, 5633, 6105, 6545, 8193, 8385

Offset: 1

N. J. A. Sloane, Feb 16 2013

Obtained by sorting and removing duplicates from the first 500 terms of A222311. There is no proof as yet that this list is complete up to 105. Only the first three terms shown are certain. Is there a proof that 4 cannot appear?

Cristian Cobeli, Mihai Prunescu, Alexandru Zaharescu, A growth model based on the arithmetic Z-game, arXiv:1511.04315 [math.NT], 2015.
Cristian Cobeli, Alexandru Zaharescu, A game with divisors and absolute differences of exponents, arXiv:1411.1334 [math.NT], 2014 (see page 12).

Cf. A222310, A222311.

terms = 40; nmax0 = 5000;
seq[nmax_] := seq[nmax] = Union[Print[nmax]; Join[r = {1}, Table[Reverse[r = FoldList[#1*(#2/GCD[#1, #2]^2) & , n, r]], {n, 2, nmax}][[All, 1]]]][[1 ;; terms]];
seq[nmax = nmax0]; seq[nmax = 2 nmax]; While[seq[nmax] == seq[nmax/2], nmax = 2 nmax]; seq[nmax] (* Jean-François Alcover, Sep 04 2018, after Ivan Neretin in A222310 *)

Corrected and extended using data from Cobeli et al., 2015. - N. J. A. Sloane, Aug 27 2016

More terms (computed from a list of 10000) from Jean-François Alcover, Sep 04 2018

cp's OEIS Frontend

A222311 Leading diagonal of triangle in A222310.

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A036262 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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A222313 A222311 sorted and duplicates removed (conjectured).

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