A112543 -id:A112543 - cp's OEIS frontend

A167192 Triangle read by rows: T(n,k) = (n-k)/gcd(n,k), 1 <= k <= n.

0, 1, 0, 2, 1, 0, 3, 1, 1, 0, 4, 3, 2, 1, 0, 5, 2, 1, 1, 1, 0, 6, 5, 4, 3, 2, 1, 0, 7, 3, 5, 1, 3, 1, 1, 0, 8, 7, 2, 5, 4, 1, 2, 1, 0, 9, 4, 7, 3, 1, 2, 3, 1, 1, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 11, 5, 3, 2, 7, 1, 5, 1, 1, 1, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13, 6, 11, 5, 9, 4, 1, 3, 5, 2, 3

Offset: 1

Reinhard Zumkeller, Oct 30 2009

			The triangle T(n,k) begins:
n\k   1   2   3   4  5  6  7  8  9 10 11 12 13  14  15 ...
1:    0
2:    1   0
3:    2   1   0
4:    3   1   1   0
5:    4   3   2   1  0
6:    5   2   1   1  1  0
7:    6   5   4   3  2  1  0
8:    7   3   5   1  3  1  1  0
9:    8   7   2   5  4  1  2  1  0
10:   9   4   7   3  1  2  3  1  1  0
11:  10   9   8   7  6  5  4  3  2  1  0
12:  11   5   3   2  7  1  5  1  1  1  1  0
13:  12  11  10   9  8  7  6  5  4  3  2  1  0
14:  13   6  11   5  9  4  1  3  5  2  3  1  1   0
15:  14  13   4  11  2  3  8  7  2  1  4  1  2   1   0
- _Wolfdieter Lang_, Feb 20 2013

Indranil Ghosh, Rows 1..120 of triangle, flattened

Cf. A054531, A112543, A164306.

Flatten[Table[(n-k)/GCD[n,k],{n,20},{k,n}]] (* Harvey P. Dale, Nov 27 2015 *)

for(n=1,10, for(k=1,n, print1((n-k)/gcd(n,k), ", "))) \\ G. C. Greubel, Sep 13 2017

T(n,k) = (n-k)/gcd(n,k), 1 <= k <= n.
T(n,k) = A025581(n,k)/A050873(n,k);
T(n,1) = A001477(n-1);
T(n,2) = A026741(n-2) for n > 1;
T(n,3) = A051176(n-3) for n > 2;
T(n,4) = A060819(n-4) for n > 4;
T(n,n-3) = A144437(n) for n > 3;
T(n,n-2) = A000034(n) for n > 2;
T(n,n-1) = A000012(n);
T(n,n) = A000004(n).

A112544 Denominators of fractions n/k in array by antidiagonals.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 2, 3, 4, 1, 1, 1, 2, 5, 1, 2, 3, 4, 5, 6, 1, 1, 3, 1, 5, 3, 7, 1, 2, 1, 4, 5, 2, 7, 8, 1, 1, 3, 2, 1, 3, 7, 4, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 1, 1, 1, 5, 1, 7, 2, 3, 5, 11, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 1, 3, 2, 5, 3, 1, 4, 9, 5, 11, 6, 13, 1, 2, 1, 4, 1, 2, 7, 8, 3, 2, 11, 4, 13, 14

Offset: 1

Franklin T. Adams-Watters, Sep 11 2005

			Array begins as:
  1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...;
  1, 1, 3, 2, 5, 3, 7, 4, 9,  5, ...;
  1, 2, 1, 4, 5, 2, 7, 8, 3, 10, ...;
  1, 1, 3, 1, 5, 3, 7, 2, 9,  5, ...;
  1, 2, 3, 4, 1, 6, 7, 8, 9,  2, ...;
  1, 1, 1, 2, 5, 1, 7, 4, 3,  5, ...;
  1, 2, 3, 4, 5, 6, 1, 8, 9, 10, ...;
  1, 1, 3, 1, 5, 3, 7, 1, 9,  5, ...;
  1, 2, 1, 4, 5, 2, 7, 8, 1, 10, ...;
  1, 1, 3, 2, 1, 3, 7, 4, 9,  1, ...;
Antidiagonal triangle begins as:
  1;
  1, 2;
  1, 1, 3;
  1, 2, 3, 4;
  1, 1, 1, 2, 5;
  1, 2, 3, 4, 5, 6;
  1, 1, 3, 1, 5, 3, 7;
  1, 2, 1, 4, 5, 2, 7, 8;
  1, 1, 3, 2, 1, 3, 7, 4, 9;
  1, 2, 3, 4, 5, 6, 7, 8, 9, 10;

G. C. Greubel, Antidiagonals n = 1..50, flattened

Numerators in A112543. See comments and references there.

Cf. A332049.

[Denominator((n-k+1)/k): k in [1..n], n in [1..20]]; // G. C. Greubel, Jan 12 2022

Table[Denominator[(n-k+1)/k], {n,20}, {k,n}]//Flatten (* G. C. Greubel, Jan 12 2022 *)

t1(n) = binomial(floor(3/2+sqrt(2*n)),2) -n+1;
t2(n) = n-binomial(floor(1/2+sqrt(2*n)),2);
vector(100,n,t2(n)/gcd(t1(n),t2(n)))

flatten([[denominator((n-k+1)/k) for k in (1..n)] for n in (1..20)]) # G. C. Greubel, Jan 12 2022

From G. C. Greubel, Jan 12 2022: (Start)
A(n, k) = denominator(n/k) (array).
T(n, k) = denominator((n-k+1)/k) (antidiagonal triangle).
Sum_{k=1..n} T(n, k) = A332049(n+1).
T(n, k) = A112543(n, n-k). (End)

Keyword tabl added by Franklin T. Adams-Watters, Sep 02 2009

A270861 Irregular triangle read by rows: numerators of the coefficients of polynomials J(2n-1,z) = Sum_(k=1,2, .. n) ((n+1)^2 - k + (n+1-k)z^n)z^(k-1)/k.

Original entry on oeis.org

3, 1, 8, 7, 2, 1, 15, 7, 13, 3, 1, 1, 24, 23, 22, 21, 4, 3, 2, 1, 35, 17, 11, 8, 31, 5, 2, 1, 1, 1, 48, 47, 46, 45, 44, 43, 6, 5, 4, 3, 2, 1, 63, 31, 61, 15, 59, 29, 57, 7, 3, 5, 1, 3, 1, 1, 80, 79, 26, 77, 76, 25, 74, 73, 8, 7, 2, 5, 4, 1, 2, 1

Offset: 1

Paul Curtz, Mar 24 2016

Irregular triangle of fractions:
3,     1,
8,   7/2,    2,  1/2,
15,    7, 13/3,    3,    1,  1/3,
24, 23/2, 22/3, 21/4,    4,  3/2, 2/3, 1/4,
35,   17,   11,    8, 31/5,    5,   2,   1, 1/2, 1/5,
48, 47/2, 46/3, 45/4, 44/5, 43/6,   6, 5/2, 4/3, 3/4, 2/5, 1/6.
etc.
First column: A005563; T(n, 1) = A005563(n).
Main diagonal: T(n, n) - n = n^2+1 = A002522(n).
The first upper diagonal is T(n, n+1) = n.
Consider TT(n, k) = k*T(n, k) for k = 1 to n:
3,
8, 7,
15, 14, 13,
24, 23, 22, 21,
etc.
Row sums: 3, 8+7, ... , are the positive terms of A059270; that is A059270(n).

			Irregular triangle:
3,   1,
8,   7,  2,  1,
15,  7, 13,  3,  1,  1,
24, 23, 22, 21,  4,  3, 2, 1,
35, 17, 11,  8, 31,  5, 2, 1, 1, 1
48, 47, 46, 45, 44, 43, 6, 5, 4, 3, 2, 1
etc.
Second half part by row: A112543.

Jean-François Alcover, Roots of J(2n-1,z) lie close to two concentric circles (example)

Cf. A002260, A002378, A002522, A005563, A059270, A112543, A122197, A004736.

row[n_] := CoefficientList[Sum[(((n + 1)^2 - k + (n + 1 - k)*z^n))*z^(k - 1)/k, {k, n}], z]; Table[row[n] // Numerator, {n, 1, 9}] // Flatten (* Jean-François Alcover, Apr 07 2016 *)

A374739 a(1) = 1, a(2) = 4; for n > 2, a(n) is the smallest unused positive number that shares a factor with a(n-1) while a(n)/gcd(a(n),a(n-1)) does not equal any previous term.

Original entry on oeis.org

1, 4, 6, 9, 15, 10, 14, 16, 12, 8, 20, 22, 26, 34, 36, 21, 33, 39, 51, 54, 30, 18, 27, 45, 25, 35, 49, 77, 55, 65, 85, 95, 38, 46, 48, 28, 42, 57, 69, 72, 40, 24, 44, 52, 58, 62, 64, 56, 68, 74, 82, 86, 94, 100, 60, 50, 70, 91, 119, 133, 161, 115, 145, 87, 93, 96, 66, 78, 102, 106, 118, 122, 126

Offset: 1

Scott R. Shannon, Jul 18 2024

The sequence shows long runs of both even and odd terms; in the first 100000 terms the longest run of even terms is 979 while the longest run of odd terms is 3668. In the same range the vast majority of terms with a(n) > n are odd; only 914 even terms are above this line while 60073 odd terms are, the majority of the later having only two prime factors - see the linked image.
Unless the sequence starts with primes no other primes can appear in the sequence, hence is natural to start the sequence with a(1) = 1 and a(2) = 4.
The fixed points begin 1, 25, 548, 1617, 2763, 3897, 5253, although it is likely there are many more.

			a(3) = 6 as 6 shares a factor with a(2) = 4 and 6/gcd(6,4) = 3, and 3 does not equal any previous term.
a(10) = 8 as 8 shares a factor with a(9) = 12 and 8/gcd(8,12) = 2, and 2 does not equal any previous term.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first 100000 terms. Numbers with two, three, four, or five and more prime factors, counted with multiplicity, are show as yellow, green, blue and violet respectively. The white line is a(n) = n.

Cf. A064413, A003989, A112543.

cp's OEIS Frontend

A167192 Triangle read by rows: T(n,k) = (n-k)/gcd(n,k), 1 <= k <= n.

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A112544 Denominators of fractions n/k in array by antidiagonals.

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A270861 Irregular triangle read by rows: numerators of the coefficients of polynomials J(2n-1,z) = Sum_(k=1,2, .. n) ((n+1)^2 - k + (n+1-k)*z^n)*z^(k-1)/k.

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A374739 a(1) = 1, a(2) = 4; for n > 2, a(n) is the smallest unused positive number that shares a factor with a(n-1) while a(n)/gcd(a(n),a(n-1)) does not equal any previous term.

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A270861 Irregular triangle read by rows: numerators of the coefficients of polynomials J(2n-1,z) = Sum_(k=1,2, .. n) ((n+1)^2 - k + (n+1-k)z^n)z^(k-1)/k.