A363003 -id:A363003 - cp's OEIS frontend

A080839 Number of positive increasing integer sequences of length n with Gilbreath transform (that is, the diagonal of leading successive absolute differences) given by {1,1,1,1,1,...}.

1, 1, 1, 2, 6, 27, 180, 1786, 26094, 559127, 17535396, 804131875, 53833201737

Offset: 1

John W. Layman, Mar 28 2003

From T. D. Noe, Feb 05 2007: (Start)
The slowest-growing sequence of length n is 1,2,4,6,...,2(n-1). The fastest-growing sequence is 1,2,4,8,...,2^(n-1).
The ratio a(n+1)a(n-1)/a(n)^2 appears to converge to a constant near 1.46, which is the approximate growth rate of A001609. Are the sequences related?
(End)
Also, a(n) is the number of (not necessarily increasing) positive integer sequences of length n-1 with Gilbreath transform (1, ..., 1). - Pontus von Brömssen, May 13 2023

			The table below shows that {1,2,4,6,10} is one of the 6 sequences of length 5 that satisfy the stated condition:
   1
   2 1
   4 2 1
   6 2 0 1
  10 4 2 2 1

Cf. A001609, A036262, A363002, A363003, A363004, A363005.

Cf. also A136465, the total number of increasing sequences with the same maximum length. [From Charles R Greathouse IV, Aug 08 2010]

More terms from T. D. Noe, Feb 05 2007

Added "positive" to definition. - N. J. A. Sloane, May 13 2023

A363002 Number of positive nondecreasing integer sequences of length n whose Gilbreath transform is (1, 1, ..., 1).

Original entry on oeis.org

1, 1, 1, 2, 5, 17, 82, 573, 5839, 86921, 1890317, 60013894, 2778068147

Offset: 0

Pontus von Brömssen, May 13 2023

			For n = 4, the a(4) = 5 sequences are:
  (1, 2, 2, 2),
  (1, 2, 2, 4),
  (1, 2, 4, 4),
  (1, 2, 4, 6),
  (1, 2, 4, 8).

Cf. A080839 (increasing sequences), A362451, A363003, A363004 (distinct positive integers), A363005 (distinct integers).

A363004 Number of sequences of n distinct positive integers whose Gilbreath transform is (1, 1, ..., 1).

Original entry on oeis.org

1, 1, 1, 1, 2, 7, 40, 355, 4819, 99242, 3049155, 138762035

Offset: 0

Pontus von Brömssen, May 13 2023

			For n = 5, the a(5) = 7 sequences are:
  (1, 2, 4, 6,  8),
  (1, 2, 4, 6, 10),
  (1, 2, 4, 8,  6),
  (1, 2, 4, 8, 10),
  (1, 2, 4, 8, 12),
  (1, 2, 4, 8, 14),
  (1, 2, 4, 8, 16).

Cf. A080839 (increasing sequences), A363002 (nondecreasing sequences), A363003, A363005 (distinct integers).

A363005 Number of sequences of n distinct integers whose Gilbreath transform is (1, 1, ..., 1).

Original entry on oeis.org

1, 1, 2, 4, 12, 56, 416, 4764, 84272, 2278740, 92890636, 5659487836

Offset: 0

Pontus von Brömssen, May 13 2023

a(n) is even for all n >= 2, because if the sequence (x_1, ..., x_n) has Gilbreath transform (1, ..., 1), so has the sequence (2 - x_1, ..., 2 - x_n).

Negative terms are permitted.

			For n = 4, the following 6 sequences, together with the sequences obtained by replacing each term x by 2-x in each of these sequences, have Gilbreath transform (1, 1, 1, 1), so a(4) = 12.
  (1, 2, 0, -4),
  (1, 2, 0, -2),
  (1, 2, 0,  4),
  (1, 2, 4,  0),
  (1, 2, 4,  6),
  (1, 2, 4,  8).

Cf. A080839 (increasing sequences), A363002 (nondecreasing sequences), A363003, A363004 (distinct positive integers).

A366032 Difference d between the least odd integer that would disprove Gilbreath's conjecture and prime(n).

Original entry on oeis.org

2, 4, 2, 10, 6, 10, 6, 6, 12, 16, 16, 16, 8, 12, 10, 30, 20, 26, 34, 20, 28, 18, 26, 30, 36, 24, 28, 26, 30, 88, 54, 68, 44, 64, 46, 46, 48, 40, 36, 52, 32, 64, 46, 66, 36, 66, 94, 72, 66, 76, 60, 54, 56, 70, 58, 66, 74, 72, 76, 56, 84, 80, 88, 70, 92, 104, 78, 86, 100, 84, 66, 86, 84, 86, 96

Offset: 3

Giorgos Kalogeropoulos, Sep 27 2023

nonn

In Gilbreath's conjecture the leading row lists the primes. In this sequence we take as leading row the first n-1 primes joined with the least odd integer k that disproves Gilbreath's conjecture instead of prime(n).
The terms of the sequence are the difference of this hypothetical number k and prime(n).
k is always greater than prime(n-1). The first 1000 terms show that k is greater than prime(n).
Although the first 1000 terms are positive, in theory a term can be negative: prime(n-1) < k < prime(n).
If we find a term that is zero then k = prime(n) and that would disprove the conjecture.

			The first term of the sequence is a(3) = 2 (offset is 3)
We start with the first 2 primes and instead of the third prime, we choose k=7.
  2,3  -->  2,3,7  instead of  2,3,5
  1         1,4                1,2
            3                  1
.
k=7 is the least odd integer that disproves the conjecture. So, a(3) = k-prime(3) = 7 - 5 = 2.
.
  2,3,5,7,11  -->  2,3,5,7,11,23  instead of  2,3,5,7,11,13
  1,2,2,4          1,2,2,4,12                 1,2,2,4,2
  1,0,2            1,0,2,8                    1,0,2,2
  1,2              1,2,6                      1,2,0
  1                1,4                        1,2
                   3                          1
k=23 is the least odd integer that disproves the conjecture. So, a(6) = k-prime(6) = 23 - 13 = 10.

Cf. A036262, A363003.

Table[(k=Prime@n;While[Nest[Abs@*Differences,Join[Prime@Range@n,{k}],n]=={1},k=k+2];k)-NextPrime@Prime@n,{n,2,100}]

isok(v) = my(nb=#v); for (i=1, nb-1, v = vector(#v-1, k, abs(v[k+1]-v[k]));); v[1] == 1;
a(n) = my(v = primes(n-1), k=prime(n)); while (isok(concat(v, k)), k+=2); k - prime(n); \\ Michel Marcus, Sep 28 2023

cp's OEIS Frontend

A080839 Number of positive increasing integer sequences of length n with Gilbreath transform (that is, the diagonal of leading successive absolute differences) given by {1,1,1,1,1,...}.

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A363002 Number of positive nondecreasing integer sequences of length n whose Gilbreath transform is (1, 1, ..., 1).

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A363004 Number of sequences of n distinct positive integers whose Gilbreath transform is (1, 1, ..., 1).

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A363005 Number of sequences of n distinct integers whose Gilbreath transform is (1, 1, ..., 1).

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A366032 Difference d between the least odd integer that would disprove Gilbreath's conjecture and prime(n).

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