A362451 Gilbreath transform of {sigma(i), i >= 1} (cf. A000203).

1, 2, 1, 1, 1, 2, 1, 0, 0, 1, 0, 4, 0, 3, 0, 2, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 68, 0, 14, 0, 7, 0, 2, 0, 21, 1, 8, 1, 9, 1, 0, 1, 18, 0, 7, 0, 2, 0, 1, 0, 13, 1, 1, 1, 2, 1, 1

Offset: 1

Author

N. J. A. Sloane, May 03 2023

Keywords

• nonn

Comments

Given a sequence {u(i), i >= o} with offset o, its absolute difference sequence is the sequence {v(i) = |u(i+1)-u(i)|, i >= o}.

The Gilbreath transform of a sequence s = {s(i), i >= o} is constructed as follows.

Form an array A in which the initial row is s and each subsequence row is the absolute difference sequence of the previous row. The sequence of leading terms of the rows of A is the Gilbreath transform of s.

If "absolute difference sequence" is changed to the familiar "first differences", instead of the Gilbreath transform we get the usual inverse binomial transform.

It appears that the terms are mostly 0's and 1's, with occasional eruptions of "geysers". See A362456, A362457.

Examples

			We give two examples. (1) For the Gilbreath transform of the sequence of primes (cf. A000040), the array A is given in A036262. The Gilbreath transform begins {2, 1, 1, 1, 1, ...}, and the famous Gilbreath conjecture is that every term after the initial 2 is equal to 1.
(2) For the Gilbreath transform of {tau(i), i >= 1} (cf. A000005), the array A is given in A362450, and the Gilbreath transform is given in A361897. The authors of the latter sequence conjecture that its terms are just 0's and 1's.
See A362452 for a further example.
		

Links

• N. J. A. Sloane, Table of n, a(n) for n = 1..60000
• N. J. A. Sloane, Maple code for Gilbreath transform and related arrays
• N. J. A. Sloane, Transforms
• 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.)
• Paolo Xausa, Table of n, a(n) for n = 1..1000000
• Paolo Xausa, Logarithmic scatterplot for n = 1..1100000
• Index entries for sequences related to Gilbreath conjecture and transform

Crossrefs

Cf. A000005, A000040, A000203, A036262, A361897, A362450, A362452, A362456, A362457, A362464.

Programs

Maple

# To get M terms of the Gilbreath transform of s, assuming offset is 1:
GT := proc(s,M) local G,u,i;
u := [seq(s[i],i=1..M)];
G:=[s[1]];
for i from 1 to M-1 do
u:=[seq(abs(u[i+1]-u[i]),i=1..nops(u)-1)];
G:=[op(G),u[1]]; od:
G;
end;
# For the present sequence:
GT(numtheory[sigma],150);
# See link for a more comprehensive Maple program

Mathematica

A362451[nmax_]:=Module[{d=DivisorSigma[1,Range[nmax]]},Join[{1},Table[First[d=Abs[Differences[d]]],nmax-1]]];A362451[200] (* Paolo Xausa, May 07 2023 *)

PARI

lista(nn) = my(v=apply(sigma, [1..nn]), list = List(), nb=nn); listput(list, v[1]); for (n=2, nn, nb--; my(w = vector(nb, k, abs(v[k+1]-v[k]))); listput(list, w[1]); v = w; ); Vec(list);
lista(200) \\ (based on PARI program in A361897)

Extensions

More than the usual number of terms are displayed in order to go out beyond the long initial 0,1 subsequence.