A126988 -id:A126988 - cp's OEIS frontend

A123229 Triangle read by rows: T(n, m) = n - (n mod m).

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

Offset: 1

Roger L. Bagula and Gary W. Adamson, Oct 06 2006

An equivalent definition: Consider A000012 as a lower-left all-1's triangle, and build the matrix product by multiplication with A127093 from the right. That is, T(n,m) = Sum_{j=m..n} A000012(n,j)*A127093(j,m) = Sum_{j=m..n} A127093(j,m) = m*floor(n/m) = m*A010766(n,m). - Gary W. Adamson, Jan 05 2007

The number of parts k in the triangle is A000203(k) hence the sum of parts k is A064987(k). - Omar E. Pol, Jul 05 2014

			Triangle begins:
{1},
{2, 2},
{3, 2, 3},
{4, 4, 3, 4},
{5, 4, 3, 4, 5},
{6, 6, 6, 4, 5, 6},
{7, 6, 6, 4, 5, 6, 7},
{8, 8, 6, 8, 5, 6, 7, 8},
{9, 8, 9, 8, 5, 6, 7, 8, 9},
...

G. C. Greubel, Rows n=1..100 of triangle, flattened

Cf. A074025, A093960.

Cf. also A024916 (row sums), A127093, A127094, A127096, A127097, A127098, A127099, A038040, A000203, A126988, A127013, A127057.

Flat(List([1..10],n->List([1..n],m->n-(n mod m)))); # Muniru A Asiru, Oct 12 2018

seq(seq(n-modp(n,m),m=1..n),n=1..13); # Muniru A Asiru, Oct 12 2018

a = Table[Table[n - Mod[n, m], {m, 1, n}], {n, 1, 20}]; Flatten[a]

for(n=1,9,for(m=1,n,print1(n-n%m", "))) \\ Charles R Greathouse IV, Nov 07 2011

Edited by N. J. A. Sloane, Jul 05 2014 at the suggestion of Omar E. Pol, who observed that A127095 (Gary W. Adamson, with edits by R. J. Mathar) was the same as this sequence.

A127094 Triangle, reversal of A127093.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Jan 05 2007

			First few rows of the triangle are:
  1;
  2, 1;
  3, 0, 1;
  4, 0, 2, 1;
  5, 0, 0, 0, 1;
  6, 0, 0, 3, 2, 1;
  ...

G. C. Greubel, Rows n = 1..30 of the triangle, flattened

Cf. A000203 (row sums), A126988, A127093.

[n mod (k-n-1) - (n+1) mod (k-n-1) + 1: k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 08 2021

Table[Mod[n, k-n-1] - Mod[n+1, k-n-1] +1, {n,12}, {k,n}]//Flatten (* G. C. Greubel, Mar 08 2021 *)

flatten([[n%(k-n-1) - (n+1)%(k-n-1) + 1 for k in [1..n]] for n in [1..12]]) # G. C. Greubel, Mar 08 2021

Reversed rows of A127093.

T(n, K) = mod(n, k-n-1) - mod(n+1, k-n-1) + 1. - Mats Granvik, Sep 02 2007

A130054 Inverse Moebius transform of A023900.

Original entry on oeis.org

1, 0, -1, -1, -3, 0, -5, -2, -3, 0, -9, 1, -11, 0, 3, -3, -15, 0, -17, 3, 5, 0, -21, 2, -7, 0, -5, 5, -27, 0, -29, -4, 9, 0, 15, 3, -35, 0, 11, 6, -39, 0, -41, 9, 9, 0, -45, 3, -11, 0, 15, 11, -51, 0, 27, 10, 17, 0, -57, -3, -59, 0, 15, -5, 33, 0, -65, 15, 21

Offset: 1

Gary W. Adamson, May 04 2007

Multiplicative because A023900 is. - Andrew Howroyd, Aug 03 2018

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Andrew Howroyd)

Row sums of A129691.

Cf. A126988, A000005, A023900, A062570, A001511, A018804, A000203, A007431.

[&+[d*MoebiusMu(d)*NumberOfDivisors(n div d):d in Divisors(n)]:n in [1..70]]; // Marius A. Burtea, Nov 17 2019

with(numtheory): seq(add(d*mobius(d)*tau(n/d), d in divisors(n)), n=1..60); # Ridouane Oudra, Nov 17 2019

b[n_] := Sum[d MoebiusMu[d], {d, Divisors[n]}];
a[n_] := Sum[b[n/d], {d, Divisors[n]}];
a /@ Range[1, 100] (* Jean-François Alcover, Sep 20 2019, from PARI *)
f[p_, e_] := 1-(p-1)*e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 23 2020 *)

\\ here b(n) is A023900
b(n)={sumdivmult(n, d, d*moebius(d))}
a(n)={sumdiv(n, d, b(n/d))} \\ Andrew Howroyd, Aug 03 2018

A126988 * A130054 = d(n), A000005: (1, 2, 2, 3, 2, 4, 2, 4, 3, 4, ...).
a(n) = Sum_{d|n} A023900(n/d). - Andrew Howroyd, Aug 03 2018
a(n) = Sum_{d|n} d*mu(d)*tau(n/d). - Ridouane Oudra, Nov 17 2019
From Werner Schulte, Sep 06 2020: (Start)
Multiplicative with a(p^e) = 1 - (p-1) * e for prime p and e >= 0.
Dirichlet g.f.: (zeta(s))^2 / zeta(s-1).
Dirichlet convolution with A062570 equals A001511.
Dirichlet convolution with A018804 equals A000203.
Dirichlet inverse of A007431. (End)
a(n) = 1 - Sum_{k=1..n-1} a(gcd(n,k)). - Ilya Gutkovskiy, Nov 06 2020

Name changed and terms a(11) and beyond from Andrew Howroyd, Aug 03 2018

A140256 Triangle read by columns: Column k is A014963 aerated with groups of (k-1) zeros.

Original entry on oeis.org

1, 2, 1, 3, 0, 1, 2, 2, 0, 1, 5, 0, 0, 0, 1, 1, 3, 2, 0, 0, 1, 7, 0, 0, 0, 0, 0, 1, 2, 2, 0, 2, 0, 0, 0, 1, 3, 0, 3, 0, 0, 0, 0, 0, 1, 1, 5, 0, 0, 2, 0, 0, 0, 0, 1, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 3, 0, 2, 0, 0, 0, 0, 0, 1, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 7, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson and Mats Granvik, May 16 2008, Jun 11 2008

If the row number n is prime, the row consists of T(n,1)=n followed by n-2 zeros and followed by T(n,n)=1.
Similar to A138618.
Row products of nonzero terms in row n, equals n. - Mats Granvik, May 22 2016

			First few rows of the triangle are:
   1;
   2, 1;
   3, 0, 1;
   2, 2, 0, 1;
   5, 0, 0, 0, 1;
   1, 3, 2, 0, 0, 1;
   7, 0, 0, 0, 0, 0, 1;
   2, 2, 0, 2, 0, 0, 0, 1;
   3, 0, 3, 0, 0, 0, 0, 0, 1;
   1, 5, 0, 0, 2, 0, 0, 0, 0, 1;
  11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
   1, 1, 2, 3, 0, 2, 0, 0, 0, 0, 0, 1;
  ...
Column 2 = (1, 0, 2, 0, 3, 0, 2, 0, 5, 0, 1, 0, 7, ...).

T. Tao, Simons Lecture I: Structure and randomness in Fourier analysis and number theory.
Wikipedia, Fundamental theorem of arithmetic.

Cf. A140255 (row sums), A014963.

Row products without the zero terms produce A000027. [Mats Granvik, Oct 08 2009]

=if(row()>=column();if(mod(row();column())=0;lookup(roundup(row()/column();0);A000027;A014963);0);"")

t[n_, k_] /; Divisible[n, k] := Exp[ MangoldtLambda[n/k] ]; t[, ] = 0; Table[t[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 28 2013 *)
(* recurrence *)
Clear[t, s, n, k, z, nn];z = 1;nn = 14;t[n_, k_] := t[n, k] = If[k == 1, Zeta[s]*(1 - 1/n^(s - 1)) -Sum[t[n, i]/i^(s - 1), {i, 2, n}], If[Mod[n, k] == 0, t[n/k, 1], 0], 0]; A = Table[Table[Limit[t[n, k], s -> z], {k, 1, n}], {n, 1, nn}]; Flatten[Exp[A]*Table[Table[If[Mod[n, k] == 0, 1, 0], {k, 1, n}], {n, 1, nn}]] (* Mats Granvik, Apr 09 2016, May 22 2016 *)

T(n,k) = A014963(n/k) = A014963(A126988(n,k)) if k|n, T(n,k)=0 otherwise. 1 <= k <= n.
From Mats Granvik, Apr 10 2016, May 22 2016: (Start)
Limit as s -> 1 of the recurrence: Ts(n, k) = if k = 1 then zeta(s)*(1 - 1/n^(s - 1)) -Sum_{i=2..n} Ts(n, i)/(i)^(s - 1) else if n mod k = 0 then Ts(n/k, 1) else 0 else 0.
For n not equal to k: Limit as s -> 1 of the recurrence: Ts(n, k) = if k = 1 then zeta(s) -Sum_{i=2..n} Ts(n, i)/i^(s - 1) else if n mod k = 0 then Ts(n/k, 1) else 0 else 0.
Limit as s -> 1 of the recurrence: Ts(n, k) = if k = 1 then log(n) -Sum_{i=2..n} Ts(n, i)/i^(s - 1) else if n mod k = 0 then Ts(n/k, 1) else 0 else 0. (End)
[The above sentences need a lot of work!  Parentheses might help. - N. J. A. Sloane, Mar 14 2017]

A127096 Triangle T(n,m) = A000012*A127094 read by rows.

Original entry on oeis.org

1, 3, 1, 6, 1, 1, 10, 1, 3, 1, 15, 1, 3, 1, 1, 21, 1, 3, 4, 3, 1, 28, 1, 3, 4, 3, 1, 1, 36, 1, 3, 4, 7, 1, 3, 1, 45, 1, 3, 4, 7, 1, 6, 1, 1, 55, 1, 3, 4, 7, 6, 6, 1, 3, 1, 66, 1, 3, 4, 7, 6, 6, 1, 3, 1, 1, 78, 1, 3, 4, 7, 6, 12, 1, 7, 4, 3, 1, 91, 1, 3, 4, 7, 6, 12, 1, 7, 4, 3, 1, 1, 105, 1, 3, 4, 7, 6, 12, 8, 7, 4, 3, 1, 3, 1

Offset: 1

Gary W. Adamson, Jan 05 2007

Consider A000012 as a lower-left all-1's triangle, and build the matrix product by multiplication with A127094 from the right.

			First few rows of the triangle are:
   1;
   3, 1,
   6, 1, 1;
  10, 1, 3, 1;
  15, 1, 3, 1, 1;
  21, 1, 3, 4, 3, 1;
  28, 1, 3, 4, 3, 1, 1;
  ...

Cf. A127093, A127094, A123229, A024916 (row sums), A000203, A126988.

A127093 := proc(n,m) if n mod m = 0 then m; else 0 ; fi; end:
A127094 := proc(n,m) A127093(n, n-m+1) ; end:
A127096 := proc(n,m) add( A127094(j,m),j=m..n) ; end:
for n from 1 to 15 do for m from 1 to n do printf("%d,",A127096(n,m)) ; od: od: # R. J. Mathar, Aug 18 2009

T[n_, m_] := Sum[1 + Mod[j, m - j - 1] - Mod[1 + j, m - j - 1], {j, m, n}];
Table[T[n, m], {n, 1, 14}, {m, 1, n}] // Flatten (* Jean-François Alcover, Sep 15 2023 *)

T(n,m) = Sum_{j=m..n} A000012(n,j)*A127094(j,m) = Sum_{j=m..n} A127094(j,m).

Edited and extended by R. J. Mathar, Aug 18 2009

A127446 Triangle T(n,k) = n*A051731(n,k) read by rows.

Original entry on oeis.org

1, 2, 2, 3, 0, 3, 4, 4, 0, 4, 5, 0, 0, 0, 5, 6, 6, 6, 0, 0, 6, 7, 0, 0, 0, 0, 0, 7, 8, 8, 0, 8, 0, 0, 0, 8, 9, 0, 9, 0, 0, 0, 0, 0, 9, 10, 10, 0, 0, 10, 0, 0, 0, 0, 10, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 12, 12, 12, 12, 0, 12, 0, 0, 0, 0, 0, 12, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 14, 14, 0, 0, 0, 0, 14

Offset: 1

Gary W. Adamson, Jan 14 2007

Replace the 1's in row n of A051731 with n's.

T(n,k) is the sum of the k's in the partitions of n into equal parts. - Omar E. Pol, Nov 25 2019

			First few rows of the triangle:
  1;
  2, 2;
  3, 0, 3;
  4, 4, 0, 4;
  5, 0, 0, 0, 5;
  6, 6, 6, 0, 0, 6;
  7, 0, 0, 0, 0, 0, 7;
  ...
For n = 6 the partitions of 6 into equal parts are [6], [3,3], [2,2,2], [1,1,1,1,1,1], so the sum of the k's are [6, 6, 6, 0, 0, 6] respectively, equaling the 6th row of triangle. - _Omar E. Pol_, Nov 25 2019

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Cf. A038040 (row sums), A051731, A126988, A244051, A328362.

a127446 n k = a127446_tabl !! (n-1) !! (k-1)
a127446_row n = a127446_tabl !! (n-1)
a127446_tabl = zipWith (\v ws -> map (* v) ws) [1..] a051731_tabl
-- Reinhard Zumkeller, Jan 21 2014

A127446 := proc(n,k) if n mod k = 0 then n; else 0; fi; end: for n from 1 to 20 do for k from 1 to n do printf("%d,",A127446(n,k)) ; od: od: # R. J. Mathar, May 08 2009

Flatten[Table[If[Mod[n, k] == 0, n, 0], {n, 20}, {k, n}]] (* Vincenzo Librandi, Nov 02 2016 *)

T(n,k) = k*A126988(n,k). - Omar E. Pol, Nov 25 2019

Edited and extended by R. J. Mathar, May 08 2009

A280499 Triangular table for division in ring GF(2)[X]: T(n,k) = n/k, or 0 if k is not a divisor of n, where the binary expansion of each number defines the corresponding (0,1)-polynomial.

Original entry on oeis.org

1, 2, 1, 3, 0, 1, 4, 2, 0, 1, 5, 0, 3, 0, 1, 6, 3, 2, 0, 0, 1, 7, 0, 0, 0, 0, 0, 1, 8, 4, 0, 2, 0, 0, 0, 1, 9, 0, 7, 0, 0, 0, 3, 0, 1, 10, 5, 6, 0, 2, 3, 0, 0, 0, 1, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 12, 6, 4, 3, 0, 2, 0, 0, 0, 0, 0, 1, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 14, 7, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 15, 0, 5, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Antti Karttunen, Jan 09 2017

This is GF(2)[X] analog of A126988, using "carryless division in base-2" instead of ordinary division.

The triangular table T(n,k), n=1.., k=1..n is read by rows: T(1,1), T(2,1), T(2,2), T(3,1), T(3,2), T(3,3), etc.

			The first 17 rows of the triangle:
   1
   2 1
   3 0 1
   4 2 0 1
   5 0 3 0 1
   6 3 2 0 0 1
   7 0 0 0 0 0 1
   8 4 0 2 0 0 0 1
   9 0 7 0 0 0 3 0 1
  10 5 6 0 2 3 0 0 0 1
  11 0 0 0 0 0 0 0 0 0 1
  12 6 4 3 0 2 0 0 0 0 0 1
  13 0 0 0 0 0 0 0 0 0 0 0 1
  14 7 0 0 0 0 2 0 0 0 0 0 0 1
  15 0 5 0 3 0 0 0 0 0 0 0 0 0 1
  16 8 0 4 0 0 0 2 0 0 0 0 0 0 0 1
  17 0 15 0 5 0 0 0 0 0 0 0 0 0 3 0 1
  -----------------------------------
7 ("111" in binary) encodes polynomial X^2 + X + 1, which is irreducible over GF(2) (7 is in A014580), so it is divisible only by itself and 1, and thus T(7,1) = 7, T(7,k) = 0 for k=2..6 and T(7,7) = 1.
9 ("1001" in binary) encodes polynomial X^3 + 1, which is factored over GF(2) as (X+1)(X^2 + X + 1), and thus T(9,3) = 7 and T(9,7) = 3 because the polynomial X + 1 is encoded by 3 ("11" in binary).

Lower triangular region of square array A280500.
Transpose: A280494.
Cf. A014580, A048720, A126988, A178908, A280500, A280493 (the row sums).

(define (A280499 n) (A280500bi (A002024 n) (A002260 n))) ;; Code for A280500bi given in A280500.

T(n,k) = the unique d such that A048720(d,k) = n, provided that such d exists, otherwise zero.

A127057 Triangle T(n,k), partial row sums of the n-th row of A127013 read right to left.

Original entry on oeis.org

1, 3, 1, 4, 1, 1, 7, 3, 1, 1, 6, 1, 1, 1, 1, 12, 6, 3, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 15, 7, 3, 3, 1, 1, 1, 1, 13, 4, 4, 1, 1, 1, 1, 1, 1, 18, 8, 3, 3, 3, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 28, 16, 10, 6, 3, 3, 1, 1, 1, 1, 1, 1, 14, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 24, 10, 3, 3, 3, 3, 3, 1, 1

Offset: 1

Gary W. Adamson, Jan 04 2007

Also partial row sums of the n-th row of A126988 read left to right. - Reinhard Zumkeller, Jan 21 2014

			The triangle starts
   1;
   3, 1;
   4, 1, 1;
   7, 3, 1, 1;
   6, 1, 1, 1, 1;
  12, 6, 3, 1, 1, 1;
   8, 1, 1, 1, 1, 1, 1;
  15, 7, 3, 3, 1, 1, 1, 1;
  13, 4, 4, 1, 1, 1, 1, 1, 1;
  18, 8, 3, 3, 3, 1, 1, 1, 1, 1; ...

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Cf. A126988, A127013, A000203, A038040.

a127057 n k = a127057_tabl !! (n-1) !! (k-1)
a127057_row n = a127057_tabl !! (n-1)
a127057_tabl = map (scanr1 (+)) a126988_tabl
-- Reinhard Zumkeller, Jan 21 2014

A126988:= func< n,k | (n mod k) eq 0 select n/k else 0 >;
T:= func< n,k | (&+[A126988(n, j): j in [k..n]]) >;
[[T(n,k): k in [1..n]]: n in [1..12]]; // G. C. Greubel, Jun 03 2019

A126988[n_, m_]:= If[Mod[n, m]==0, n/m, 0];
T[n_, m_]:= Sum[A126988[n, j], {j,m,n}];
Table[T[n, m], {n,1,12}, {m,1,n}]//Flatten (* G. C. Greubel, Jun 03 2019 *)

A126988(n, k) = if(n%k==0, n/k, 0);
T(n,k) = sum(j=k,n, A126988(n,j));
for(n=1, 12, for(k=1,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Jun 03 2019

def A126988(n, k):
    if (n%k==0): return n/k
    else: return 0
def T(n,k): return sum(A126988(n,j) for j in (k..n))
[[T(n, k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jun 03 2019

T(n,k) = Sum_{i=1..n-k+1} A127013(n,i), n>=1, 1<=k<=n.
T(n,k) = Sum_{i=k..n} A126988(n,i).
Row sums: Sum_{k=1..n} T(n,k) = A038040(n).
T(n,1) = A000203(n).
T = A126988 * M as infinite lower triangular matrices, M = (1; 1, 1; 1, 1, 1; ...).

Edited and extended by R. J. Mathar, Jul 23 2008

A127172 Cube of A051731.

Original entry on oeis.org

1, 3, 1, 3, 0, 1, 6, 3, 0, 1, 3, 0, 0, 0, 1, 9, 3, 3, 0, 0, 1, 3, 0, 0, 0, 0, 0, 1, 10, 6, 0, 3, 0, 0, 0, 1, 6, 0, 3, 0, 0, 0, 0, 0, 1, 9, 3, 0, 0, 0, 3, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Jan 06 2007

Nonzero terms in every column = A007425: (1, 3, 3, 6, 3, 9, 3, ...).
Row sums = A007426: (1, 4, 4, 20, 4, 16, ...).
A127172 * mu(n) = d(n); or A127172 * A008683 = A000005.
A127172 * d(n) = tau_5(n); or A127172 * A000005 = A061200.
A127172 * phi(n) = A007429: (1, 4, 5, 11, 7, 20, ...); or: A127172 * A000010 = A007429.
Note that A051731 * d(n) = row sums of A127172; or A051731 * A000005 = A007425.
Also, A126988 * mu(n) = phi(n); or A126988 * A008683 = A000010.
A126988 * phi(n) = A018804: (1, 3, 5, 8, 9, 15, ...); = A127170 * mu(n).

			First few rows of the triangle:
   1;
   3, 1;
   3, 0, 1;
   6, 3, 0, 1;
   3, 0, 0, 0, 1;
   9, 3, 3, 0, 0, 1;
   3, 0, 0, 0, 0, 0, 1;
  10, 6, 0, 3, 0, 0, 0, 1;
   6, 0, 3, 0, 0, 0, 0, 0, 1;
   9, 3, 0, 0, 3, 0, 0, 0, 0, 1;
  ...

Cf. A000005, A007425, A127170, A051731, A007429, A000010, A126988, A018804.

Cube of A051731 A007425: (1, 3, 3, 6, 3, 9, 3, ...) in every column k, interspersed with (k-1) zeros.

A141671 Triangle T(n, k) = n/k if n mod k = 0, otherwise T(n, k) = 0, with T(n, 0) = n+1, read by rows.

Original entry on oeis.org

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

Offset: 0

Roger L. Bagula and Gary W. Adamson and Mats Granvik, Sep 06 2008

Apparently this is different from A141672. - N. J. A. Sloane, Sep 13 2008

For n, k >= 1 this triangle is the same as A126988(n, k). - G. C. Greubel, Mar 16 2024

			Triangle begins as:
   1;
   2,  1;
   3,  2, 1;
   4,  3, 0, 1;
   5,  4, 2, 0, 1;
   6,  5, 0, 0, 0, 1;
   7,  6, 3, 2, 0, 0, 1;
   8,  7, 0, 0, 0, 0, 0, 1;
   9,  8, 4, 0, 2, 0, 0, 0, 1;
  10,  9, 0, 3, 0, 0, 0, 0, 0, 1;
  11, 10, 5, 0, 0, 2, 0, 0, 0, 0, 1;

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Cf. A126988, A141672.

A141671:= func< n,k | k eq 0 select n+1 else (n mod k) eq 0 select n/k else 0>;
[A141671(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Mar 16 2024

T[n_, k_]= If[k==0, n+1, If[Mod[n,k]==0, n/k, 0]];
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten

T(m,n)={if(m, if(n%m, 0, n/m), n+1)};
for(n=0, 10, for(m=0, n, print1(T(m,n)","))) \\ Charles R Greathouse IV, Oct 11 2009

def A141671(n, k):
    if k==0: return n+1
    elif (n%k==0): return n//k
    else: return 0
flatten([[A141671(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Mar 16 2024

T(n, k) = n/k if n mod k = 0, otherwise T(n, k) = 0, with T(n, 0) = n+1.

Edited by G. C. Greubel, Mar 16 2024

cp's OEIS Frontend

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