A075362 -id:A075362 - cp's OEIS frontend

A193649 Q-residue of the (n+1)st Fibonacci polynomial, where Q is the triangular array (t(i,j)) given by t(i,j)=1. (See Comments.)

1, 1, 3, 5, 15, 33, 91, 221, 583, 1465, 3795, 9653, 24831, 63441, 162763, 416525, 1067575, 2733673, 7003971, 17938661, 45954543, 117709185, 301527355, 772364093, 1978473511

Offset: 0

Clark Kimberling, Aug 02 2011

nonn

Suppose that p=p(0)*x^n+p(1)*x^(n-1)+...+p(n-1)*x+p(n) is a polynomial of positive degree and that Q is a sequence of polynomials: q(k,x)=t(k,0)*x^k+t(k,1)*x^(k-1)+...+t(k,k-1)*x+t(k,k), for k=0,1,2,... The Q-downstep of p is the polynomial given by D(p)=p(0)*q(n-1,x)+p(1)*q(n-2,x)+...+p(n-1)*q(0,x)+p(n).

Since degree(D(p))

Example: let p(x)=2*x^3+3*x^2+4*x+5 and q(k,x)=(x+1)^k.

D(p)=2(x+1)^2+3(x+1)+4(1)+5=2x^2+7x+14

D(D(p))=2(x+1)+7(1)+14=2x+23

D(D(D(p)))=2(1)+23=25;

the Q-residue of p is 25.

We may regard the sequence Q of polynomials as the triangular array formed by coefficients:

t(0,0)

t(1,0)....t(1,1)

t(2,0)....t(2,1)....t(2,2)

t(3,0)....t(3,1)....t(3,2)....t(3,3)

and regard p as the vector (p(0),p(1),...,p(n)). If P is a sequence of polynomials [or triangular array having (row n)=(p(0),p(1),...,p(n))], then the Q-residues of the polynomials form a numerical sequence.

Following are examples in which Q is the triangle given by t(i,j)=1 for 0<=i<=j:

Q.....P...................Q-residue of P

1.....1...................A000079, 2^n

1....(x+1)^n..............A007051, (1+3^n)/2

1....(x+2)^n..............A034478, (1+5^n)/2

1....(x+3)^n..............A034494, (1+7^n)/2

1....(2x+1)^n.............A007582

1....(3x+1)^n.............A081186

1....(2x+3)^n.............A081342

1....(3x+2)^n.............A081336

1.....A040310.............A193649

1....(x+1)^n+(x-1)^n)/2...A122983

1....(x+2)(x+1)^(n-1).....A057198

1....(1,2,3,4,...,n)......A002064

1....(1,1,2,3,4,...,n)....A048495

1....(n,n+1,...,2n).......A087323

1....(n+1,n+2,...,2n+1)...A099035

1....p(n,k)=(2^(n-k))*3^k.A085350

1....p(n,k)=(3^(n-k))*2^k.A090040

1....A008288 (Delannoy)...A193653

1....A054142..............A101265

1....cyclotomic...........A193650

1....(x+1)(x+2)...(x+n)...A193651

1....A114525..............A193662

More examples:

Q...........P.............Q-residue of P

(x+1)^n...(x+1)^n.........A000110, Bell numbers

(x+1)^n...(x+2)^n.........A126390

(x+2)^n...(x+1)^n.........A028361

(x+2)^n...(x+2)^n.........A126443

(x+1)^n.....1.............A005001

(x+2)^n.....1.............A193660

A094727.....1.............A193657

(k+1).....(k+1)...........A001906 (even-ind. Fib. nos.)

(k+1).....(x+1)^n.........A112091

(x+1)^n...(k+1)...........A029761

(k+1)......A049310........A193663

(In these last four, (k+1) represents the triangle t(n,k)=k+1, 0<=k<=n.)

A051162...(x+1)^n.........A193658

A094727...(x+1)^n.........A193659

A049310...(x+1)^n.........A193664

A075362....A075362........A193665

Changing the notation slightly leads to the Mathematica program below and the following formulation for the Q-downstep of p: first, write t(n,k) as q(n,k). Define r(k)=Sum{q(k-1,i)*r(k-1-i) : i=0,1,...,k-1} Then row n of D(p) is given by v(n)=Sum{p(n,k)*r(n-k) : k=0,1,...,n}.

			First five rows of Q, coefficients of Fibonacci polynomials (A049310):
1
1...0
1...0...1
1...0...2...0
1...0...3...0...1
To obtain a(4)=15, downstep four times:
D(x^4+3*x^2+1)=(x^3+x^2+x+1)+3(x+1)+1: (1,1,4,5) [coefficients]
DD(x^4+3*x^2+1)=D(1,1,4,5)=(1,2,11)
DDD(x^4+3*x^2+1)=D(1,2,11)=(1,14)
DDDD(x^4+3*x^2+1)=D(1,14)=15.

Cf. A192872 (polynomial reduction), A193091 (polynomial augmentation), A193722 (the upstep operation and fusion of polynomial sequences or triangular arrays).

q[n_, k_] := 1;
r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}];
f[n_, x_] := Fibonacci[n + 1, x];
p[n_, k_] := Coefficient[f[n, x], x, k]; (* A049310 *)
v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]
Table[v[n], {n, 0, 24}]    (* A193649 *)
TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
Table[r[k], {k, 0, 8}]  (* 2^k *)
TableForm[Table[p[n, k], {n, 0, 6}, {k, 0, n}]]

Conjecture: G.f.: -(1+x)*(2*x-1) / ( (x-1)*(4*x^2+x-1) ). - R. J. Mathar, Feb 19 2015

A023855 a(n) = 1(n) + 2(n-1) + 3(n-2) + ... + (n+1-k)k, where k = floor((n+1)/2).

Original entry on oeis.org

1, 2, 7, 10, 22, 28, 50, 60, 95, 110, 161, 182, 252, 280, 372, 408, 525, 570, 715, 770, 946, 1012, 1222, 1300, 1547, 1638, 1925, 2030, 2360, 2480, 2856, 2992, 3417, 3570, 4047, 4218, 4750, 4940, 5530, 5740, 6391, 6622, 7337, 7590, 8372, 8648, 9500, 9800, 10725, 11050

Offset: 1

Clark Kimberling

Given a rectangle of perimeter 2*n one can form rectangles having this perimeter for a number of different rectangles or squares depending on how large 2*n is. The sequence lists the total areas of all such rectangles for each 2*n. - J. M. Bergot, Sep 14 2011

Antidiagonal sums of triangle A075362. - L. Edson Jeffery, Jan 20 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1)

Cf. A000292, A023856, A023857, A024305, A024854.

a023855 n = sum $ zipWith (*) [1 .. div (n+1) 2] [n, n-1 ..]
-- Reinhard Zumkeller, Jan 23 2012

[(4*n^3 +15*n^2 +14*n +3 -3*(n+1)^2*(-1)^n)/48: n in [1..60]]; // G. C. Greubel, Jul 12 2022

seq(-(1/3)*floor((k+1)/2)^3 + (k/2)*floor((k+1)/2)^2 + ((3*k+2)/6)*floor((k+1)/2), k=1..100); # Wesley Ivan Hurt, Sep 18 2013

LinearRecurrence[{1,3,-3,-3,3,1,-1}, {1,2,7,10,22,28,50}, 60] (* Vincenzo Librandi, Jan 23 2012 *)
Table[-Ceiling[n/2] (Ceiling[n/2] + 1) (2 Ceiling[n/2] - 3 n - 2)/6, {n, 100}] (* Wesley Ivan Hurt, Sep 20 2013 *)

a(n)=if(n%2, (n+1)*(n+3)*(2*n+1)/24, n*(n+1)*(n+2)/12)

my(x='x+O('x^99)); Vec(x*(1+x+2*x^2)/((1-x)^4*(1+x)^3)) \\ Altug Alkan, Mar 03 2018

[(4*n^3 +15*n^2 +14*n +3 -3*(n+1)^2*(-1)^n)/48 for n in (1..60)] # G. C. Greubel, Jul 12 2022

a(n) = (n+1)*(n+3)*(2*n+1)/24 if n is odd, or n*(n+1)*(n+2)/12 if n is even.
G.f.: x*(1+x+2*x^2)/((1-x)^4*(1+x)^3). - Ralf Stephan, Apr 28 2004
a(n) = Sum_{i=1..ceiling(n/2)} i*(n-i+1) = -ceiling(n/2)*(ceiling(n/2)+1)*(2*ceiling(n/2)-3n-2)/6. - Wesley Ivan Hurt, Sep 19 2013
a(n) = (4*n^3 + 15*n^2 + 14*n + 3 - 3*(n+1)^2*(-1)^n)/48. - Luce ETIENNE, Oct 22 2014
a(n) = (A000292(n) + (n mod 2)*(ceiling(n/2))^2)/2. - Luc Rousseau, Feb 25 2018
E.g.f.: (1/24)*( x*(21+12*x+2*x^2)*cosh(x) + (3+12*x+15*x^2+2*x^3)*sinh(x) ). - G. C. Greubel, Jul 12 2022

Formula, program, and slight revision by Charles R Greathouse IV, Feb 23 2010

A051173 Triangle read by rows: T(n, k) = lcm(n, k).

Original entry on oeis.org

1, 2, 2, 3, 6, 3, 4, 4, 12, 4, 5, 10, 15, 20, 5, 6, 6, 6, 12, 30, 6, 7, 14, 21, 28, 35, 42, 7, 8, 8, 24, 8, 40, 24, 56, 8, 9, 18, 9, 36, 45, 18, 63, 72, 9, 10, 10, 30, 20, 10, 30, 70, 40, 90, 10, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 11, 12, 12, 12, 12, 60, 12, 84, 24, 36, 60, 132, 12

Offset: 1

Clark Kimberling

			Triangle begins (for the full array see A109042):
  [1]  1;
  [2]  2,  2;
  [3]  3,  6,  3;
  [4]  4,  4, 12,  4;
  [5]  5, 10, 15, 20,  5;
  [6]  6,  6,  6, 12, 30,  6;
  [7]  7, 14, 21, 28, 35, 42,  7;
  [8]  8,  8, 24,  8, 40, 24, 56,  8;

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened
Keith Bourque and Steve Ligh, Matrices associated with multiplicative functions, Linear Algebra and its Applications 216 (1995) 267-275.
Shaofang Hong and Raphael Loewy, Asymptotic behavior of eigenvalues of greatest common divisor matrices, Glasgow Mathematical Journal, Vol. 46, No. 3 (2004) 551-569.
Eric Weisstein's World of Mathematics, Least Common Multiple
Wikipedia, Least Common Multiple
Index entries for sequences related to lcm's

Cf. A109043 (column 2), A051193 (row sums), A000384 (central terms).

Cf. A109042 (variant), A003990, A002260, A075362, A051537, A050873.

a051173 = lcm
a051173_row n = a051173_tabl !! (n-1)
a051173_tabl = map (\x -> map (lcm x) [1..x]) [1..]
-- Reinhard Zumkeller, Aug 13 2013, Jul 07 2013

A051173 := proc(u,v) ilcm(u,v) ; end proc: # R. J. Mathar, Apr 07 2011

Table[LCM[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 30 2018 *)

T(n,k) = lcm(n,k);
tabl(nn) = for (n=1, nn, for (k=1, n, print1(T(n,k), ", ")); print;) \\ Michel Marcus, Jul 10 2017

T(n, 1) = T(n, n) = n. T(n, 2) = A109043(n). - R. J. Mathar, Apr 07 2011
T(n, k) = A075362(n, k)/A050873(n, k), 1 <= k <= n. - Reinhard Zumkeller, Apr 25 2011
T(n, k) = A051537(n, k) * A050873(n, k). - Reinhard Zumkeller, Jul 07 2013

A066680 Badly sieved numbers: as in the Sieve of Eratosthenes multiples of unmarked numbers p are marked, but only up to p^2.

Original entry on oeis.org

2, 3, 5, 7, 8, 11, 12, 13, 17, 18, 19, 23, 27, 29, 30, 31, 37, 41, 43, 45, 47, 50, 53, 59, 61, 63, 67, 70, 71, 73, 75, 79, 80, 83, 89, 97, 98, 101, 103, 105, 107, 109, 112, 113, 125, 127, 128, 131, 137, 139, 147, 149, 151, 154, 157, 163

Offset: 1

Reinhard Zumkeller, Dec 31 2001

A099104(a(n)) = 1.
a(A207432(n)) = A000040(n). - Reinhard Zumkeller, Feb 17 2012
Obviously all primes and cubes of primes are in the sequence, while squares of primes are not. In fact, A000225 tells us which exponents prime powers in the sequence will exhibit.
But where it gets really interesting is in what happens to the Achilles numbers: the smallest badly sieved numbers that are also Achilles numbers are 864 and 972. - Alonso del Arte, Feb 21 2012
From Peter Munn, Aug 09 2019: (Start)
The factorization pattern of a number's divisors (as defined in A191743) determines whether a number is a term.
There are no semiprimes in the sequence, and a 3-almost prime is present if and only if its largest prime factor is less than its square root. The first term that is a 4-almost prime is 220.
The effect of this sieve can be compared against the A270877 trapezoidal sieve. Each unmarked number k marks k-1 numbers in both sieves; but the largest number marked by k in this sieve is k^2, about twice the largest number marked by k in A270877 (the triangular number T_k = k(k+1)/2). The relative densities early in the two sequences are illustrated by a(10) = 18 < A270877(10) = 19, a(100) = 312 > A270877(100) = 268, a(1000) = 4297 > A270877(1000) = 2894. (End)

			For 2, the first unmarked number, there is only one multiple <= 4=2^2:
giving 2 3 [4] 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ...
for 3, the next unmarked number, we mark 6=2*3 and 9=3*3
giving 2 3 [4] 5 [6] 7 8 [9] 10 11 12 13 14 15 16 17 18 19 20 ...
for 5, the next unmarked number, we mark 10=2*5, 15=3*5, 20=4*5 and 25=5*5
giving 2 3 [4] 5 [6] 7 8 [9] [10] 11 12 13 14 [15] 16 17 18 19 [20] ... and so on.

T. D. Noe, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Sieve
Wikipedia, Sieve theory
Index entries for sequences generated by sieves

A066681, A066682, A066683, A099042, A099043, A207432 have analysis of this sequence.
Cf. A056875, A075362, A099104 (characteristic function), A191743.
Sequences generated by a closely related sieving process: A000040 (also a subsequence), A026424, A270877.

a066680 n = a066680_list !! (n-1)
a066680_list = s [2..] where
   s (b:bs) = b : s [x | x <- bs, x > b ^ 2 || mod x b > 0]
-- Reinhard Zumkeller, Feb 17 2012

A099104[1] = 0; A099104[n_] := A099104[n] = Product[If[n > d^2, 1, 1 - A099104[d]], {d, Select[ Range[n-1], Mod[n, #] == 0 &]}]; Select[ Range[200], A099104[#] == 1 &] (* Jean-François Alcover, Feb 15 2012 *)
max = 200; badPrimes = Range[2, max]; len = max; iter = 1; While[iter <= len, curr = badPrimes[[iter]]; badPrimes = Complement[badPrimes, Range[2, curr]curr]; len = Length[badPrimes]; iter++]; badPrimes (* Alonso del Arte, Feb 21 2012 *)

A215631 Triangle read by rows: T(n,k) = n^2 + n*k + k^2, 1 <= k <= n.

Original entry on oeis.org

3, 7, 12, 13, 19, 27, 21, 28, 37, 48, 31, 39, 49, 61, 75, 43, 52, 63, 76, 91, 108, 57, 67, 79, 93, 109, 127, 147, 73, 84, 97, 112, 129, 148, 169, 192, 91, 103, 117, 133, 151, 171, 193, 217, 243, 111, 124, 139, 156, 175, 196, 219, 244, 271, 300, 133, 147, 163

Offset: 1

Reinhard Zumkeller, Nov 11 2012

			The triangle begins:
row n   T(n,k), 1 <= k <= n
   1:     3
   2:     7   12
   3:    13   19   27
   4:    21   28   37   48
   5:    31   39   49   61   75
   6:    43   52   63   76   91  108
   7:    57   67   79   93  109  127  147
   8:    73   84   97  112  129  148  169  192
   9:    91  103  117  133  151  171  193  217  243
  10:   111  124  139  156  175  196  219  244  271  300
  11:   133  147  163  181  201  223  247  273  301  331  363
  12:   157  172  189  208  229  252  277  304  333  364  397  432

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened

Cf. A215646 (row sums), A002061 (left edge, shifted), A033428 (right edge), A003215.

a215631 n k = a215631_tabl !! (n-1) !! (k-1)
a215631_row n = a215631_tabl !! (n-1)
a215631_tabl = zipWith3 (zipWith3 (\u v w -> u + v + w))
                        a093995_tabl a075362_tabl a133819_tabl
-- Reinhard Zumkeller, Nov 11 2012

[[i^2+i*j+j^2: j in [1..i]]: i in [1..10]]; // Vincenzo Librandi, Jun 07 2015

seq(seq(i^2+i*j+j^2, j=1..i),i=1..10); # Robert Israel, May 10 2015

Table[n^2 + n*k + k^2, {n, 11}, {k, n}] // Flatten (* Michael De Vlieger, May 12 2015 *)

for(n=1,15,for(k=1,n,print1(n^2+n*k+k^2,", "))) \\ Derek Orr, May 13 2015

T(n,k) = 2*A070216(n,k) - A215630(n,k).
G.f. for triangle: (3-2*x+3*x*y+x^2-11*x^2*y+4*x^3*y+x^3*y^2+x^4*y^2)*x*y/((1-x)^3*(1-x*y)^3). - Robert Israel, May 10 2015
From Avi Friedlich, May 26 2015: (Start)
T(n,k) = A093995(n,k) + A075362(n,k) + A133819(n,k).
T(k+1,k) = A003215(k).
T(k+2,k) = A003215(k)/2 + A003215(k+1)/2.
T(k+3,k) = A003215(k)/3 + A003215(k+1)/3 + A003215(k+2)/3 and so on. (End)

A215630 Triangle read by rows: T(n,k) = n^2 - n*k + k^2, 1 <= k <= n.

Original entry on oeis.org

1, 3, 4, 7, 7, 9, 13, 12, 13, 16, 21, 19, 19, 21, 25, 31, 28, 27, 28, 31, 36, 43, 39, 37, 37, 39, 43, 49, 57, 52, 49, 48, 49, 52, 57, 64, 73, 67, 63, 61, 61, 63, 67, 73, 81, 91, 84, 79, 76, 75, 76, 79, 84, 91, 100, 111, 103, 97, 93, 91, 91, 93, 97, 103, 111

Offset: 1

Reinhard Zumkeller, Nov 11 2012

T(n,k) = A093995(n,k) - A075362(n,k) + A133819(n,k) = 2*A070216(n,k) - A215631(n,k), 1 <= k <= n.

			The triangle begins:
.  1:     1
.  2:     3    4
.  3:     7    7    9
.  4:    13   12   13   16
.  5:    21   19   19   21   25
.  6:    31   28   27   28   31   36
.  7:    43   39   37   37   39   43   49
.  8:    57   52   49   48   49   52   57   64
.  9:    73   67   63   61   61   63   67   73   81
. 10:    91   84   79   76   75   76   79   84   91  100
. 11:   111  103   97   93   91   91   93   97  103  111  121
. 12:   133  124  117  112  109  108  109  112  117  124  133  144 .

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened

Cf. A004068 (row sums), A002061 (left edge), A000290 (right edge).

Cf. A003215 (central terms).

a215630 n k = a215630_tabl !! (n-1) !! (k-1)
a215630_row n = a215630_tabl !! (n-1)
a215630_tabl = zipWith3 (zipWith3 (\u v w -> u - v + w))
                        a093995_tabl a075362_tabl a133819_tabl

A079904 Triangle read by rows: T(n, k) = n*k, 0 <= k <= n.

Original entry on oeis.org

0, 0, 1, 0, 2, 4, 0, 3, 6, 9, 0, 4, 8, 12, 16, 0, 5, 10, 15, 20, 25, 0, 6, 12, 18, 24, 30, 36, 0, 7, 14, 21, 28, 35, 42, 49, 0, 8, 16, 24, 32, 40, 48, 56, 64, 0, 9, 18, 27, 36, 45, 54, 63, 72, 81, 0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 0, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121

Offset: 0

Reinhard Zumkeller, Feb 21 2003

See the comment in A025581 on a problem posed by François Viète (Vieta) 1593, where this triangle is related to A025581 and A257238. - Wolfdieter Lang, May 12 2015

			Triangle T(n, k) begins:
  n\k 0  1  2  3  4  5  6  7  8  9  10 ...
  0:  0
  1:  0  1
  2:  0  2  4
  3:  0  3  6  9
  4:  0  4  8 12 16
  5:  0  5 10 15 20 25
  6:  0  6 12 18 24 30 36
  7:  0  7 14 21 28 35 42 49
  8:  0  8 16 24 32 40 48 56 64
  9:  0  9 18 27 36 45 54 63 72 81
  10: 0 10 20 30 40 50 60 70 80 90 100
  ... - _Wolfdieter Lang_, May 12 2015

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).

Cf. A005843, A008585, A008586, A005563, A002378, A000290.
Cf. A075362 (without column k=0), A002411 (row sums), A001105 (central terms).
Cf. A257238, A025581.

seq(seq(n*k, k=0..n), n=0..10); # Robert Israel, May 12 2015

Array[Range[0, #^2, #] &, 15, 0] (* Paolo Xausa, Mar 27 2025 *)

row(n) = vector(n+1, i, (i-1)*n); \\ Amiram Eldar, May 12 2025

T(n, k) = n*k, 0 <= k <= n.
T(n, k) = if k = 0 then 0 else T(n, k-1) + n.
T(n, 0) = 1. T(n, 1) = n for n > 0.
T(n, 2) = A005843(n) for n > 1.
T(n, 3) = A008585(n) for n > 2.
T(n, 4) = A008586(n) for n > 3.
T(n, n-2) = A005563(n-1) for n > 1.
T(n, n-1) = A002378(n-1) for n > 0.
T(n, n) = A000290(n).
T(n, k) = (A257238(n, k) - A025581(n, k)^3) / (3*A025581(n, k)). See the Viète comment above. - Wolfdieter Lang, May 12 2015
From Robert Israel, May 12 2015: (Start)
G.f. as triangle: (1 + x*y - 2*x^2*y)*x*y/((1 - x)^2*(1 - x*y)^3).
G.f. as sequence: -(Sum_{n>=0} (n^2 - n)*x^(n*(n + 1)/2)) / (1 - x) + (Sum_{n>=1} x^(n*(n + 1)/2)) * x/(1 - x)^2. These sums are related to Jacobi Theta functions. (End)
T(n, k) = gcd(n, k) * lcm(n, k). - Peter Luschny, Mar 26 2025

A340792 List in which n appears ceiling(d(n)/2) = A038548(n) times, where d(n) is the number of divisors of n.

Original entry on oeis.org

1, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 9, 10, 10, 11, 12, 12, 12, 13, 14, 14, 15, 15, 16, 16, 16, 17, 18, 18, 18, 19, 20, 20, 20, 21, 21, 22, 22, 23, 24, 24, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 28, 29, 30, 30, 30, 30, 31, 32, 32, 32, 33, 33, 34, 34, 35, 35, 36, 36, 36, 36, 36

Offset: 1

Charles Kusniec, Jan 21 2021

The numbers in A075362 arranged in numerical order.

			Array begins:
   1
   2
   3
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Charles Kusniec, Bijection between OEIS sequence A340792 and the positive Triangular Multiplication Table

Cf. A038548 (row lengths), A075362, A161906, A340791, A061017 (comparable array).

row(n) = my(d=divisors(n), r1=select(x->(x<=sqrt(n)), d), r2=Vecrev(select(x->(x>=sqrt(n)), d))); vector(#r1, k, r1[k]*r2[k]); \\ Michel Marcus, Jan 22 2021

T(n,k) = A161906(n,k) * A340791(n,k).

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cp's OEIS Frontend

A254605 The minimum absolute difference between k*m1 and m2 (m1A075362.

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A050873 Triangular array T read by rows: T(n,k) = gcd(n,k).

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A193649 Q-residue of the (n+1)st Fibonacci polynomial, where Q is the triangular array (t(i,j)) given by t(i,j)=1. (See Comments.)

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A023855 a(n) = 1*(n) + 2*(n-1) + 3*(n-2) + ... + (n+1-k)*k, where k = floor((n+1)/2).

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A051173 Triangle read by rows: T(n, k) = lcm(n, k).

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A066680 Badly sieved numbers: as in the Sieve of Eratosthenes multiples of unmarked numbers p are marked, but only up to p^2.

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A215631 Triangle read by rows: T(n,k) = n^2 + n*k + k^2, 1 <= k <= n.

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A023855 a(n) = 1(n) + 2(n-1) + 3(n-2) + ... + (n+1-k)k, where k = floor((n+1)/2).