A368312 -id:A368312 - cp's OEIS frontend

A368103 a(1)=1; for n>1, a(n) is the smallest number not already used which has a factor difference in common with a(n-1).

1, 4, 9, 16, 7, 27, 40, 10, 18, 8, 3, 15, 24, 6, 2, 12, 5, 21, 32, 45, 13, 28, 54, 26, 42, 20, 30, 14, 36, 17, 57, 80, 35, 48, 23, 75, 11, 39, 56, 72, 22, 46, 94, 144, 19, 63, 88, 43, 135, 55, 91, 112, 25, 49, 64, 31, 99, 120, 38, 60, 29, 93, 128, 33, 65, 84, 41, 129, 176, 50, 66, 92, 141, 192

Offset: 1

Neal Gersh Tolunsky, Dec 11 2023

nonn

A factor difference of x is any abs(p-q) where x=p*q (in other words, the difference of a factor pair of x, per A368312).

Prime numbers are among the numbers which appear most delayed in this sequence. - Thomas Scheuerle, Dec 12 2023

			For n=2, a(1)=1 can be factored only as 1*1, which has difference 0. The next term cannot be 2 and 3 as they do not have a factor difference 0, but 4 = 2*2 does, so that a(2) = 4.
For n=5, a(4)=16 has factor differences 0,6,15 and the smallest unused number with one of those differences is a(5) = 7 = 7*1 difference 6.

Neal Gersh Tolunsky, Table of n, a(n) for n = 1..10000
Thomas Scheuerle, MATLAB Script.
Thomas Scheuerle, Plot of the first 393 prime numbers over their indices of appearance. It is remarkable that prime numbers do not appear in order.

Cf. A368312.

Cf. A368059 (with factor sums), A359035, A360995.

MATLAB
```
% See Scheuerle link.
```

A368571 Triangle read by rows where T(n,k) is the number of positive integers M which have both n and k as factor differences, 1 <= k < n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 2, 1, 1, 0, 0, 2, 1, 1, 0, 1, 0, 0, 1, 1, 2, 1, 1, 1, 0, 0, 2, 1, 1, 1, 2, 0, 1, 0, 0, 3, 2, 1, 3, 1, 1, 1, 1, 0, 0, 1, 1, 3, 1, 1, 1, 1, 0, 1, 0, 0, 3, 3, 1, 2, 2, 1, 2, 2, 1, 1, 0, 0, 3, 2, 1, 2, 2, 1, 2, 1, 1, 0, 1, 0, 0

Offset: 2

Kevin Ryde, Dec 30 2023

The factor differences of some M are all abs(p-q) where M = p*q for positive integers p,q, being row M of A368312.
Erdős and Rosenfeld (proposition 3.1) show that T(n,k) is finite.
Their method shows the relevant M are those M = (d^2 + (G/d)^2 - 2*(n^2+k^2))/16 which are positive integers, for G = n^2 - k^2, d < sqrt(G), and d divides G.
Diagonal T(n,n-1) = 0 since in that case M <= 0 for all d.
Diagonal T(n,n-2) = 0 since in that case M is not an integer for d=1 and otherwise M <= 0.

			Triangle begins:
      k=1  2  3  4  5  6  7  8
  n=2:  0
  n=3:  0, 0
  n=4:  1, 0, 0
  n=5:  1, 1, 0, 0
  n=6:  1, 0, 1, 0, 0
  n=7:  1, 2, 1, 1, 0, 0
  n=8:  2, 1, 1, 0, 1, 0, 0
  n=9:  1, 1, 2, 1, 1, 1, 0, 0

Kevin Ryde, Table of n, a(n) for rows n=2..150, flattened
Paul Erdős and Moshe Rosenfeld, The factor-difference set of integers, Acta Arithmetica, volume 79, number 4, 1997, pages 353-359.

Cf. A368312 (factor differences).

T(n,k) = my(t=2*(n^2+k^2), v=apply(sqr,divisors(n^2-k^2))); sum(i=1,#v\2, my(m=v[i]+v[#v-i+1]-t); m>0 && m%16==0);

T(n,k) = number of rows of A368312 which contain both n and k.

A385176 Positive half of inverse speed permutation array. Square array A(n,k), n >= 0, k >= 0, read by ascending antidiagonals.

Original entry on oeis.org

1, -1, 2, 2, 2, 3, 2, -1, 3, 4, -2, 3, 3, 4, 5, -2, 3, -1, 4, 5, 6, 3, 3, 4, 4, 5, 6, 7, 3, -2, 4, -1, 5, 6, 7, 8, 3, -2, 4, 5, 5, 6, 7, 8, 9, -3, 4, 4, 5, -1, 6, 7, 8, 9, 10, -3, 4, -2, 5, 6, 6, 7, 8, 9, 10, 11, -3, 4, -2, 5, 6, -1, 7, 8, 9, 10, 11, 12, 4, 4, 5, 5, 6, 7, 7, 8, 9, 10, 11, 12, 13

Offset: 0

Peter Munn, Jun 20 2025

Particles labeled with nonzero integers j start at time t = 0 at x = 2k (offset from the origin) on a straight line. Each particle, j, moves at speed -1/j, so crosses the origin at time t = 2j^2. T(n,k) gives the label of the particle in the line segment (2k, 2k+2) at time t = 2n+1.
It is easy to determine that particles labeled i and -j cross at x = 2*(i-j) at time t = 2ij, and that (for t > 0) a particle crosses x = 2k only when encountering a particle heading in the opposite direction. So at t = 2n+1 there is exactly one particle in each segment (2k, 2k+2) and the particle labels define a bi-infinite permution of the nonzero integers. For the terms of this sequence, we restrict k >= 0; and taking the absolute values of the terms in each row gives a permutation of the positive integers. Moreover, the differences between row n-1 and row n consist of exchanges of paired divisors of -n.
The halved positions, k, at which particles encounter a segment boundary x = 2k at t = 2n are given by row n of A368312. So when that row starts with a 0, this indicates a particle crossing the origin. On the other hand, the nonzero terms, k, of row t of A211343 indicate the segment midpoints x = 2k-1 that are encountered by particles at time t, with terms in odd (respectively even) columns corresponding to positive-labeled (respectively negative-labeled) particles.

			Square array A(n,k) begins:
   n   t\k|   0    1    2    3    4    5    6    7    8    9   10   11   12
  --------+-----------------------------------------------------------------
   0   1  |   1,   2,   3,   4,   5,   6,   7,   8,   9,  10,  11,  12,  13
   1   3  |  -1,   2,   3,   4,   5,   6,   7,   8,   9,  10,  11,  12,  13
   2   5  |   2,  -1,   3,   4,   5,   6,   7,   8,   9,  10,  11,  12,  13
   3   7  |   2,   3,  -1,   4,   5,   6,   7,   8,   9,  10,  11,  12,  13
   4   9  |  -2,   3,   4,  -1,   5,   6,   7,   8,   9,  10,  11,  12,  13
   5  11  |  -2,   3,   4,   5,  -1,   6,   7,   8,   9,  10,  11,  12,  13
   6  13  |   3,  -2,   4,   5,   6,  -1,   7,   8,   9,  10,  11,  12,  13
   7  15  |   3,  -2,   4,   5,   6,   7,  -1,   8,   9,  10,  11,  12,  13
   8  17  |   3,   4,  -2,   5,   6,   7,   8,  -1,   9,  10,  11,  12,  13
   9  19  |  -3,   4,  -2,   5,   6,   7,   8,   9,  -1,  10,  11,  12,  13
  10  21  |  -3,   4,   5,  -2,   6,   7,   8,   9,  10,  -1,  11,  12,  13
  11  23  |  -3,   4,   5,  -2,   6,   7,   8,   9,  10,  11,  -1,  12,  13
  12  25  |   4,  -3,   5,   6,  -2,   7,   8,   9,  10,  11,  12,  -1,  13

Cf. A211343, A368312.

cp's OEIS Frontend

A368103 a(1)=1; for n>1, a(n) is the smallest number not already used which has a factor difference in common with a(n-1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

MATLAB

A368571 Triangle read by rows where T(n,k) is the number of positive integers M which have both n and k as factor differences, 1 <= k < n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A385176 Positive half of inverse speed permutation array. Square array A(n,k), n >= 0, k >= 0, read by ascending antidiagonals.

Views

Author

Keywords

Comments

Examples

Crossrefs