A362450 -id:A362450 - cp's OEIS frontend

A361897 Leading terms of the rows of the array in A362450; or, Gilbreath transform of tau (A000005).

1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0

Offset: 1

Wayman Eduardo Luy and Robert G. Wilson v, Mar 28 2023

Conjecture: All terms are either 0 or 1. Verified to a(10^7).
Inspired by Gilbreath's conjecture, A036262.
Using the terminology of A362451, this is the Gilbreath transform of tau (A000005). - N. J. A. Sloane, May 05 2023

			Table begins (conjecture is leading terms are 0 or 1):
1 2 2 3 2 4 2 4 3 4 2 6 2 4 4 5 2 6 2 6 4 4 2 8 3 4 4 6 2 8 2 6 4 4 4 9 2 4 4 ...
 1 0 1 1 2 2 2 1 1 2 4 4 2 0 1 3 4 4 4 2 0 2 6 5 1 0 2 4 6 6 4 2 0 0 5 7 2 0 ...
  1 1 0 1 0 0 1 0 1 2 0 2 2 1 2 1 0 0 2 2 2 4 1 4 1 2 2 2 0 2 2 2 0 5 2 5 2 4 ...
   0 1 1 1 0 1 1 1 1 2 2 0 1 1 1 1 0 2 0 0 2 3 3 3 1 0 0 2 2 0 0 2 5 3 3 3 2 ...
    1 0 0 1 1 0 0 0 1 0 2 1 0 0 0 1 2 2 0 2 1 0 0 2 1 0 2 0 2 0 2 3 2 0 0 1 0 ...
     1 0 1 0 1 0 0 1 1 2 1 1 0 0 1 1 0 2 2 1 1 0 2 1 1 2 2 2 2 2 1 1 2 0 1 1 ...
      1 1 1 1 1 0 1 0 1 1 0 1 0 1 0 1 2 0 1 0 1 2 1 0 1 0 0 0 0 1 0 1 2 1 0 1 ...
       0 0 0 0 1 1 1 1 0 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 ...
        0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 ...
         0 0 1 1 0 0 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 ...
          0 1 0 1 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 1 0 1 1 0 1 0 0 0 0 0 1 ...
           1 1 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 1 1 1 0 1 1 1 0 0 0 0 1 ...
etc.
...
The first two rows are A000005, abs(A051950). The full table, read by antidiagonals, is A362450.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
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.)
Index entries for sequences related to Gilbreath conjecture and transform

Cf. A000005, A036262, A051950, A362450, A362451, A362452, A362453 (0's), A362454 (1's).

See also A001659 (if don't use absolute values).

N:= 200: # for a(1) to a(N)
L:= [seq(numtheory:-tau(n),n=1..N)]:
for i from 1 to 105 do
  R[i]:= L[1];
  L:= map(abs,L[2..-1]-L[1..-2])
od:
seq(R[i],i=1..M); # Robert Israel, May 07 2023

a[n_] := NestWhile[ Abs@ Differences@ # &, Table[ DivisorSigma[0, m], {m, n}], Length[##] > 1 &][[1]]; Array[a, 105]
(* or *)
mx = 105; lst = {}; k = 0; d = Array[ DivisorSigma[0, #] &, mx]; While[k < mx, AppendTo[lst, d[[1]]]; d = Abs@ Differences@ d; k++]; lst
(* or *)
A361897[nmax_]:=Module[{d=DivisorSigma[0,Range[nmax]]},Join[{1},Table[First[d=Abs[Differences[d]]],nmax-1]]];A361897[200] (* Paolo Xausa, May 07 2023 *)

lista(nn) = my(v=apply(numdiv, [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) \\ Michel Marcus, Mar 29 2023

Edited by N. J. A. Sloane, Apr 30 2023

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

Original entry on oeis.org

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

N. J. A. Sloane, May 03 2023

nonn

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.

			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.

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

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

# 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

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 *)

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)

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

A362452 Gilbreath transform of {sigma(i)-i, i >= 1} (see sum of aliquot parts, A001065).

Original entry on oeis.org

0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 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, 62, 0, 12, 0, 3, 0, 2, 0, 25, 1

Offset: 1

N. J. A. Sloane, May 03 2023

nonn

See A362451 for further information.
The first 50000 terms of the present sequence suggest that the terms are usually 0's and 1's, except for occasional "geysers". See A362458, A362459.
[It would be nice to have plots of larger numbers of initial terms.]

N. J. A. Sloane, Table of n, a(n) for n = 1..50000
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..1000000
Index entries for sequences related to Gilbreath conjecture and transform

Cf. A000005, A000040, A000203, A001065, A036262, A361897, A362450, A362451, A362458, A362459.

# To get M terms of the Gilbreath transform of s:
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:
aliq := proc(n) numtheory[sigma](n) - n; end;
GT(aliq,150);

A362452[nmax_]:=Module[{d=DivisorSigma[1,Range[nmax]]-Range[nmax]},Join[{0},Table[First[d=Abs[Differences[d]]],nmax-1]]];A362452[200] (* Paolo Xausa, May 07 2023 *)

f(n) = sigma(n) - n
lista(nn) = my(v=apply(f, [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)

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

A362453 Indices of 0's in A361897.

Original entry on oeis.org

4, 8, 9, 10, 11, 13, 14, 15, 16, 20, 24, 28, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 68, 72, 73, 74, 75, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 89, 90, 91, 93, 94, 95, 96, 100, 104, 105, 106, 107, 109, 110, 111, 112, 116, 120, 121, 122, 123, 125, 126, 127, 132, 136, 137

Offset: 1

N. J. A. Sloane, May 05 2023

nonn

Paolo Xausa, Table of n, a(n) for n = 1..20000 (terms 1..5250 from N. J. A. Sloane)

Cf. A361897, A362450, A362454.

A362453[upto_]:=Module[{d=DivisorSigma[0,Range[upto]]},Table[If[First[d=Abs[Differences[d]]]==0,n,Nothing],{n,2,upto}]];A362453[200] (* Paolo Xausa, May 07 2023 *)

A362454 Indices of 1's in A361897.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 12, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 48, 65, 66, 67, 69, 70, 71, 76, 84, 88, 92, 97, 98, 99, 101, 102, 103, 108, 113, 114, 115, 117, 118, 119, 124, 128, 129, 130, 131, 133, 134, 135, 140, 144, 145, 146, 147, 149, 150, 151, 153, 154, 155, 157, 158

Offset: 1

N. J. A. Sloane, May 05 2023

nonn

Paolo Xausa, Table of n, a(n) for n = 1..20000 (terms 1..4750 from N. J. A. Sloane)

Cf. A361897, A362450, A362453.

A362454[upto_]:=Module[{d=DivisorSigma[0,Range[upto]]},Join[{1},Table[If[First[d=Abs[Differences[d]]]==1,n,Nothing],{n,2,upto}]]];A362454[200] (* Paolo Xausa, May 07 2023 *)

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A361897 Leading terms of the rows of the array in A362450; or, Gilbreath transform of tau (A000005).

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

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A362452 Gilbreath transform of {sigma(i)-i, i >= 1} (see sum of aliquot parts, A001065).

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A362453 Indices of 0's in A361897.

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A362454 Indices of 1's in A361897.

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