This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A036262 #94 Apr 21 2025 16:03:22
%S A036262 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,
%T A036262 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,
%U A036262 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
%N 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.
%C A036262 The conjecture is that the leading term is always 1.
%C A036262 Odlyzko has checked it for primes up to pi(10^13) = 3*10^11.
%C A036262 From _M. F. Hasler_, Jun 02 2012: (Start)
%C A036262 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.
%C A036262 The previous statement can also be formulated as: Gilbreath's conjecture is equivalent to: A036277(n) > A213014(n)+2 for all n.
%C A036262 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.
%C A036262 (End)
%C A036262 From _Clark Kimberling_, Nov 27 2022: (Start)
%C A036262 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.
%C A036262 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)
%D A036262 R. K. Guy, Unsolved Problems Number Theory, A10.
%D A036262 H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Amer. Math. Soc., 1996, p. 208.
%D A036262 Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 192.
%D A036262 W. Sierpiński, L'induction incomplète dans la théorie des nombres, Scripta Math. 28 (1967), 5-13.
%D A036262 C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 410.
%H A036262 T. D. Noe, <a href="/A036262/b036262.txt">Table of n, a(n) for n = 0..5049</a>
%H A036262 R. K. Guy, <a href="/A005165/a005165.pdf">The strong law of small numbers</a>. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
%H A036262 R. B. Killgrove and K. E. Ralston, <a href="http://dx.doi.org/10.1090/S0025-5718-59-99262-2">On a conjecture concerning the primes</a>, Math.Tables Aids Comput. 13(1959), 121-122.
%H A036262 A. M. Odlyzko, <a href="http://dx.doi.org/10.1090/S0025-5718-1993-1182247-7">Iterated absolute values of differences of consecutive primes</a>, Math. Comp. 61 (1993), 373-380.
%H A036262 F. Proth, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN598948236_0004&DMDID=DMDLOG_0076&IDDOC=630831">Sur la série des nombres premiers</a>, Nouv. Corresp. Math., 4 (1878) 236-240.
%H A036262 W. Sierpiński, <a href="http://gdz.sub.uni-goettingen.de/en/dms/load/img/?PPN=PPN311570321_0013&DMDID=dmdlog6">L'induction incomplète dans la théorie des nombres</a>, Bulletin de la Société des mathématiciens et physiciens de la R.P de Serbie, Vol XIII, 1-2 (1961), Beograd, Yougoslavie.
%H A036262 N. J. A. Sloane, <a href="http://neilsloane.com/doc/sg.txt">My favorite integer sequences</a>, in Sequences and their Applications (Proceedings of SETA '98).
%H A036262 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: <a href="https://vimeo.com/866583736?share=copy">Video</a>, <a href="http://neilsloane.com/doc/EMSep2023.pdf">Slides</a>, <a href="http://neilsloane.com/doc/EMSep2023.Updates.txt">Updates</a>. (Mentions this sequence.)
%H A036262 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GilbreathsConjecture.html">Gilbreath's Conjecture</a>
%H A036262 <a href="/index/Ge#Gilbreath">Index entries for sequences related to Gilbreath conjecture and transform</a>
%F A036262 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
%e A036262 The array begins (conjecture is leading term is always 1):
%e A036262 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
%e A036262 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
%e A036262 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
%e A036262 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
%e A036262 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
%e A036262 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
%e A036262 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
%e A036262 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
%e A036262 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
%e A036262 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
%e A036262 ...
%p A036262 A036262 := proc(n, k)
%p A036262 option remember ;
%p A036262 if n = 0 then
%p A036262 ithprime(k) ;
%p A036262 else
%p A036262 abs(procname(n-1, k+1)-procname(n-1, k)) ;
%p A036262 end if;
%p A036262 end proc:
%p A036262 seq(seq( A036262(d-k,k),k=1..d),d=1..13) ; # _R. J. Mathar_, May 10 2023
%t A036262 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 *)
%o A036262 (Haskell)
%o A036262 a036262 n k = delta !! (n - k) !! (k - 1) where delta = iterate
%o A036262 (\pds -> zipWith (\x y -> abs (x - y)) (tail pds) pds) a000040_list
%o A036262 -- _Reinhard Zumkeller_, Jan 23 2011
%Y A036262 Cf. A001223, A036261, A036277, A054977, A222310, A358691, A089582 (2nd col).
%Y A036262 See A255483 for an interesting generalization.
%K A036262 tabl,easy,nice,nonn
%O A036262 0,1
%A A036262 _N. J. A. Sloane_
%E A036262 More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 23 2003
%E A036262 Definition edited by _N. J. A. Sloane_, May 03 2023