A003990 -id:A003990 - cp's OEIS frontend

A285724 Square array read by descending antidiagonals: If n > k, A(n,k) = T(lcm(n,k), gcd(n,k)), otherwise A(n,k) = T(gcd(n,k), lcm(n,k)), where T(n,k) is sequence A000027 considered as a two-dimensional table.

1, 2, 3, 4, 5, 6, 7, 16, 21, 10, 11, 12, 13, 14, 15, 16, 46, 67, 78, 55, 21, 22, 23, 106, 25, 120, 27, 28, 29, 92, 31, 191, 210, 34, 105, 36, 37, 38, 211, 80, 41, 90, 231, 44, 45, 46, 154, 277, 379, 436, 465, 406, 300, 171, 55, 56, 57, 58, 59, 596, 61, 630, 63, 64, 65, 66, 67, 232, 436, 631, 781, 862, 903, 820, 666, 465, 253, 78, 79, 80, 529, 212, 991, 302, 85, 324, 1035, 230, 561, 90, 91

Offset: 1

Antti Karttunen, May 03 2017

The array is read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

			The top left 12 X 12 corner of the array:
   1,   2,   4,   7,   11,   16,   22,   29,   37,   46,   56,   67
   3,   5,  16,  12,   46,   23,   92,   38,  154,   57,  232,   80
   6,  21,  13,  67,  106,   31,  211,  277,   58,  436,  529,   94
  10,  14,  78,  25,  191,   80,  379,   59,  631,  212,  947,  109
  15,  55, 120, 210,   41,  436,  596,  781,  991,   96, 1486, 1771
  21,  27,  34,  90,  465,   61,  862,  302,  193,  467, 2146,  142
  28, 105, 231, 406,  630,  903,   85, 1541, 1954, 2416, 2927, 3487
  36,  44, 300,  63,  820,  324, 1596,  113, 2557,  822, 3829,  355
  45, 171,  64, 666, 1035,  208, 2016, 2628,  145, 4006, 4852,  706
  55,  65, 465, 230,  101,  495, 2485,  860, 4095,  181, 5996, 1832
  66, 253, 561, 990, 1540, 2211, 3003, 3916, 4950, 6105,  221, 8647
  78,  90, 103, 117, 1830,  148, 3570,  375,  739, 1890, 8778,  265

Antti Karttunen, Table of n, a(n) for n = 1..10585; the first 145 antidiagonals of array
MathWorld, Pairing Function

Cf. A000124 (row 1), A000217 (column 1), A001844 (main diagonal).

Cf. A000027, A003989, A003990, A003991, A286101, A286102, A286155, A285722.

(define (A285724 n) (A285724bi (A002260 n) (A004736 n)))
(define (A285724bi row col) (if (> row col) (A000027bi (lcm row col) (gcd row col)) (A000027bi (gcd row col) (lcm row col))))
(define (A000027bi row col) (* (/ 1 2) (+ (expt (+ row col) 2) (- row) (- (* 3 col)) 2)))

If n > k, A(n,k) = T(lcm(n,k),gcd(n,k)), otherwise A(n,k) = T(gcd(n,k),lcm(n,k)), where T(n,k) is sequence A000027 considered as a two-dimensional table, that is, as a pairing function from N x N to N.

If n < k, A(n,k) = A286101(n,k), otherwise A(n,k) = A286102(n,k).

A287958 Table read by antidiagonals: T(n, k) = least recursive multiple of n and k; n > 0 and k > 0.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 4, 6, 6, 4, 5, 4, 3, 4, 5, 6, 10, 12, 12, 10, 6, 7, 6, 15, 4, 15, 6, 7, 8, 14, 6, 20, 20, 6, 14, 8, 9, 8, 21, 12, 5, 12, 21, 8, 9, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 11, 10, 9, 64, 35, 6, 35, 64, 9, 10, 11, 12, 22, 30, 36, 40, 42, 42, 40

Offset: 1

Rémy Sigrist, Jun 03 2017

We say that m is a recursive multiple of d iff d is a recursive divisor of m (as described in A282446).
More informally, the prime tower factorization of T(n, k) is the union of the prime tower factorizations of n and k (the prime tower factorization of a number is defined in A182318).
This sequence has connections with the classical LCM (A003990).
For any i > 0, j > 0 and k > 0:
- A007947(T(i, j)) = A007947(lcm(i, j)),
- T(i, j) >= 1,
- T(i, j) >= max(i, j),
- T(i, j) >= lcm(i, j),
- T(i, 1) = i,
- T(i, i) = i,
- T(i, j) = T(j, i) (the sequence is commutative),
- T(i, T(j, k)) = T(T(i, j), k) (the sequence is associative),
- T(i, i*j) >= i*j,
- if gcd(i, j) = 1 then T(i, j) = i*j.
See also A287957 for the GCD equivalent.

			Table starts:
n\k|     1   2   3   4   5   6   7   8   9  10
---+-----------------------------------------------
1  |     1   2   3   4   5   6   7   8   9  10  ...
2  |     2   2   6   4  10   6  14   8  18  10  ...
3  |     3   6   3  12  15   6  21  24   9  30  ...
4  |     4   4  12   4  20  12  28  64  36  20  ...
5  |     5  10  15  20   5  30  35  40  45  10  ...
6  |     6   6   6  12  30   6  42  24  18  30  ...
7  |     7  14  21  28  35  42   7  56  63  70  ...
8  |     8   8  24  64  40  24  56   8  72  40  ...
9  |     9  18   9  36  45  18  63  72   9  90  ...
10 |    10  10  30  20  10  30  70  40  90  10  ...
...
T(4, 8) = T(2^2, 2^3) = 2^(2*3) = 2^6 = 64.

Rémy Sigrist, First 100 antidiagonals of array, flattened
Rémy Sigrist, Illustration of the first terms

Cf. A003990, A007947, A182318, A282446, A287957.

T(n,k) = if (n*k==0, return (max(n,k))); my (g=factor(lcm(n,k))); return (prod(i=1, #g~, g[i,1]^T(valuation(n, g[i,1]), valuation(k, g[i,1]))))

A339394 Sum over all partitions of n of the LCM of the number of parts and the number of distinct parts.

Original entry on oeis.org

0, 1, 3, 6, 15, 26, 43, 81, 138, 218, 320, 514, 751, 1131, 1570, 2319, 3159, 4457, 6077, 8344, 11224, 15337, 20297, 26908, 35773, 46434, 60711, 78433, 100987, 129222, 166590, 209719, 267120, 335842, 423341, 527739, 659974, 816805, 1015990, 1251686, 1543864

Offset: 0

Alois P. Heinz, Dec 02 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A003990, A051173, A339312.

b:= proc(n, i, p, d) option remember; `if`(n=0, ilcm(p, d),
      add(b(n-i*j, i-1, p+j, d+signum(j)), j=`if`(i>1, 0..n/i, n)))
    end:
a:= n-> b(n$2, 0$2):
seq(a(n), n=0..50);

b[n_, i_, p_, d_] := b[n, i, p, d] = If[n == 0, LCM[p, d],
     Sum[b[n - i*j, i - 1, p + j, d + Sign[j]],
     {j, If[i > 1, Range[0, n/i], {n}]}]];
a[n_] := b[n, n, 0, 0];
a /@ Range[0, 50] (* Jean-François Alcover, Mar 09 2021, after Alois P. Heinz *)

A343033 Array T(n, k), n, k > 0, read by antidiagonals; a variant of lunar multiplication (A087062) based on prime exponents of numbers (see Comments section for precise definition).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 2, 5, 2, 1, 1, 5, 3, 3, 5, 1, 1, 6, 7, 4, 7, 6, 1, 1, 7, 15, 5, 5, 15, 7, 1, 1, 2, 11, 6, 11, 6, 11, 2, 1, 1, 3, 3, 7, 35, 35, 7, 3, 3, 1, 1, 10, 5, 4, 13, 30, 13, 4, 5, 10, 1, 1, 11, 21, 9, 5, 77, 77, 5, 9, 21, 11, 1

Offset: 1

Rémy Sigrist and N. J. A. Sloane, Apr 03 2021

To compute T(n, k):
- write the prime exponents of n and of k on two lines, right aligned (these lines correspond to rows of A067255 in reversed order),
- to "multiply" two prime numbers: take the smallest,
- to "add" two prime numbers: take the largest,
- for example, for T(12, 14):
                (11 7 5 3 2)
    12 -->              1 2
    14 -->        x 1 0 0 1
                  ---------
                        1 1
                      0 0
                    0 0
                + 1 1
                -----------
                  1 1 0 1 1  --> 462 = T(12, 14)
This sequence is closely related to lunar multiplication (A087062):
- for any b > 1, let S_b be the set of nonnegative integers m such that A051903(m)< b,
- there is a natural bijection f from S_b to the set of nonnegative integers:
        f(Product_{k >= 0} prime(k)^d(k)) = Sum_{k >= 0} d(k) * b^k,
- let g be the inverse of f,
- then for any numbers n and k in S_b, we have:
        T(n, k) = g(f(n) "*" f(k)) where "*" denotes lunar product in base b,
- the corresponding addition table is A003990.

			Array T(n, k) begins:
  n\k|  1   2   3   4   5    6    7   8   9   10   11   12   13   14
  ----  -  --  --  --  --  ---  ---  --  --  ---  ---  ---  ---  ---
    1|  1   1   1   1   1    1    1   1   1    1    1    1    1    1
    2|  1   2   3   2   5    6    7   2   3   10   11    6   13   14  --> A007947
    3|  1   3   5   3   7   15   11   3   5   21   13   15   17   33  --> A328915
    4|  1   2   3   4   5    6    7   4   9   10   11   12   13   14  --> A007948
    5|  1   5   7   5  11   35   13   5   7   55   17   35   19   65
    6|  1   6  15   6  35   30   77   6  15  210  143   30  221  462
    7|  1   7  11   7  13   77   17   7  11   91   19   77   23  119
    8|  1   2   3   4   5    6    7   8   9   10   11   12   13   14
    9|  1   3   5   9   7   15   11   9  25   21   13   45   17   33
   10|  1  10  21  10  55  210   91  10  21  110  187  210  247  910
   11|  1  11  13  11  17  143   19  11  13  187   23  143   29  209
   12|  1   6  15  12  35   30   77  12  45  210  143   60  221  462
   13|  1  13  17  13  19  221   23  13  17  247   29  221   31  299
   14|  1  14  33  14  65  462  119  14  33  910  209  462  299  238

Cf. A003990, A007947, A007948, A051903, A067255, A087062, A328915, A342767, A343035.

T(n,k) = { my (r=1, pp=factor(n)[,1]~, qq=factor(k)[,1]~); for (i=1, #pp, for (j=1, #qq, my (p=prime(primepi(pp[i])+primepi(qq[j])-1), v=valuation(r, p), w=min(valuation(n, pp[i]), valuation(k, qq[j]))); if (w>v, r*=p^(w-v)))); r }

T(n, k) = T(k, n).
T(n, 1) = 1.
T(n, 2) = A007947(n).
T(n, 3) = A328915(n).
T(n, 4) = A007948(n).
T(n, n) = A343035(n).
A051903(T(n, k)) = min(A051903(n), A051903(k)).

A082022 In the following square array a(i,j) = Least Common Multiple of i and j. Sequence contains the product of the terms of the n-th antidiagonal.

Original entry on oeis.org

1, 4, 18, 576, 1200, 518400, 1587600, 180633600, 1646023680, 13168189440000, 461039040000, 229442532802560000, 86553098311680000, 3753113311877529600, 834966920275488000000

Offset: 1

Amarnath Murthy, Apr 06 2003

nonn

If n is even and n+1 is prime, a(n) = n^2 * (n-1)!^2. If n is odd and >3, 2*(n+1)*a(n) is a perfect square, the root of which has the factor 1/2*n*(n-1)*((n-1)/2)!. This was proved by Lawrence Sze. - Ralf Stephan, Nov 16 2004

			1 2 3 4 5...
2 2 6 4 10...
3 6 3 12 15...
4 4 12 4 20...
5 10 15 20 5...
...
The same array in triangular form is
1
2 2
3 2 3
4 6 6 4
5 4 3 4 5
...
Sequence contains the product of the terms of the n-th row.

Cf. A006580, A003990, A082292.

Equals A001044(n) / A051190(n+1).

for(n=1,20,p=1:for(k=1,n,p=p*lcm(k,n+1-k)):print1(p","))

Prod(k=1...n, lcm(k, n+1-k)).

Corrected and extended by Ralf Stephan, Apr 08 2003

A264009 Table T(i,j) = nonnegative k at which lcm(i+k,j+k) reaches the minimum, read by antidiagonals (i>=1, j>=1).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 3, 0, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 5, 2, 2, 0, 0, 2, 2, 5, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 4, 1, 1, 0, 0, 1, 1, 4, 1, 0, 0, 0, 1, 2, 3, 0, 0, 0, 3, 2, 1, 0, 0, 0, 9, 0, 3, 0, 0, 0, 0, 0, 0, 3, 0, 9, 0

Offset: 1

Ivan Neretin, Oct 31 2015

T(i,j) = T(j,i).
T(i,j) <= |i-j|.
If i divides j or vice versa, then T(i,j) = 0.

			Let i=10, j=3. Then lcm(i,j)=30, lcm(i+1,j+1)=44, lcm(i+2,j+2)=60, lcm(i+3,j+3)=78, and lcm(i+4,j+4)=14, which is the minimum. Hence T(10,3)=T(3,10)=4.

Ivan Neretin, Table of n, a(n) for n = 1..8001
Index entries for sequences related to lcm's

Cf. A003990.

A305720 Square array T(n, k) read by antidiagonals, n > 0 and k > 0; for any prime number p, the p-adic valuation of T(n, k) is the product of the p-adic valuations of n and of k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 4, 3, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 16, 1, 2, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 8, 1, 4, 5, 4, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 9, 64, 1, 6, 1, 64, 9, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 4, 1, 8, 7, 8

Offset: 1

Rémy Sigrist, Jun 09 2018

The array T is completely multiplicative in both parameters.
For any n > 0 and prime number p, T(n, p) is the highest power of p dividing n.
For any function f associating a nonnegative value to any pair of nonnegative values and such that f(0, 0) = 0, we can build an analog of this sequence, say P_f, such that for any prime number p and any n and k > 0 with p-adic valuations i and j, the p-adic valuation of P_f(n, k) equals f(i, j):
   f(i, j)    P_f
   -------    ---
   i * j      T (this sequence)
   i + j      A003991 (product)
   abs(i-j)   A089913
   min(i, j)  A003989 (GCD)
   max(i, j)  A003990 (LCM)
   i AND j    A059895
   i OR j     A059896
   i XOR j    A059897
If log(N) denotes the set {log(n) : n is in N, the set of the positive integers}, one can define a binary operation on log(N): with prime factorizations n = Product p_i^e_i and k = Product p_i^f_i, set log(n) o log(k) = Sum_{i} (e_i*f_i) * log(p_i). o has the premises of a scalar product even if log(N) isn't a vector space. T(n, k) can be viewed as exp(log(n) o log(k)). - Luc Rousseau, Oct 11 2020

			Array T(n, k) begins:
  n\k|    1    2    3    4    5    6    7    8    9   10
  ---+--------------------------------------------------
    1|    1    1    1    1    1    1    1    1    1    1
    2|    1    2    1    4    1    2    1    8    1    2  -> A006519
    3|    1    1    3    1    1    3    1    1    9    1  -> A038500
    4|    1    4    1   16    1    4    1   64    1    4
    5|    1    1    1    1    5    1    1    1    1    5  -> A060904
    6|    1    2    3    4    1    6    1    8    9    2  -> A065331
    7|    1    1    1    1    1    1    7    1    1    1  -> A268354
    8|    1    8    1   64    1    8    1  512    1    8
    9|    1    1    9    1    1    9    1    1   81    1
   10|    1    2    1    4    5    2    1    8    1   10  -> A132741

Cf. A003989, A003990, A003991, A006519, A007947, A038500, A054496, A059895, A059896, A059897, A060904, A065331, A089913, A132741, A268354, A268357.

T[n_, k_] := With[{p = FactorInteger[GCD[n, k]][[All, 1]]}, If[p == {1}, 1, Times @@ (p^(IntegerExponent[n, p] * IntegerExponent[k, p]))]];
Table[T[n-k+1, k], {n, 1, 15}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 11 2018 *)

T(n, k) = my (p=factor(gcd(n, k))[,1]); prod(i=1, #p, p[i]^(valuation(n, p[i]) * valuation(k, p[i])))

T(n, k) = T(k, n) (T is commutative).
T(m, T(n, k)) = T(T(m, n), k) (T is associative).
T(n, k) = 1 iff gcd(n, k) = 1.
T(n, n) = A054496(n).
T(n, A007947(n)) = n.
T(n, 1) = 1.
T(n, 2) = A006519(n).
T(n, 3) = A038500(n).
T(n, 4) = A006519(n)^2.
T(n, 5) = A060904(n).
T(n, 6) = A065331(n).
T(n, 7) = A268354(n).
T(n, 8) = A006519(n)^3.
T(n, 9) = A038500(n)^2.
T(n, 10) = A132741(n).
T(n, 11) = A268357(n).

A342905 Array read by antidiagonals: T(n,k) = product of all distinct primes dividing n*k (n>=1, k>=1).

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 6, 6, 2, 5, 2, 3, 2, 5, 6, 10, 6, 6, 10, 6, 7, 6, 15, 2, 15, 6, 7, 2, 14, 6, 10, 10, 6, 14, 2, 3, 2, 21, 6, 5, 6, 21, 2, 3, 10, 6, 6, 14, 30, 30, 14, 6, 6, 10, 11, 10, 3, 2, 35, 6, 35, 2, 3, 10, 11, 6, 22, 30, 6, 10, 42, 42, 10, 6, 30, 22, 6

Offset: 1

Rémy Sigrist and N. J. A. Sloane, Apr 02 2021

			The array begins:
1, 2, 3, 2, 5, 6, 7, 2, 3, 10, ...,
2, 2, 6, 2, 10, 6, 14, 2, 6, 10, ...,
3, 6, 3, 6, 15, 6, 21, 6, 3, 30, ...,
2, 2, 6, 2, 10, 6, 14, 2, 6, 10, ...,
5, 10, 15, 10, 5, 30, 35, 10, 15, 10, ...,
6, 6, 6, 6, 30, 6, 42, 6, 6, 30, ...,
7, 14, 21, 14, 35, 42, 7, 14, 21, 70, ...,
2, 2, 6, 2, 10, 6, 14, 2, 6, 10, ...,
3, 6, 3, 6, 15, 6, 21, 6, 3, 30, ...,
10, 10, 30, 10, 10, 30, 70, 10, 30, 10, ...,
...,
The first few antidiagonals are:
1,
2, 2,
3, 2, 3,
2, 6, 6, 2,
5, 2, 3, 2, 5,
6, 10, 6, 6, 10, 6,
7, 6, 15, 2, 15, 6, 7,
2, 14, 6, 10, 10, 6, 14, 2,
3, 2, 21, 6, 5, 6, 21, 2, 3,
...

Rémy Sigrist, Table of n, a(n) for n = 1..10011

A variant of the GCD array A003989 and the LCM array A003990.

Cf. A007947.

PARI
```
T(n, k) = vecprod(factor(n*k)[,1]~)
```

T(n, k) = A007947(n * k).

A374352 a(n) = [n>1] * a(n-1) + Sum_{d|n} phi(lcm(d,n/d)) where [] is an Iverson bracket.

Original entry on oeis.org

1, 3, 7, 12, 20, 28, 40, 52, 66, 82, 102, 122, 146, 170, 202, 228, 260, 288, 324, 364, 412, 452, 496, 544, 588, 636, 684, 744, 800, 864, 924, 980, 1060, 1124, 1220, 1290, 1362, 1434, 1530, 1626, 1706, 1802, 1886, 1986, 2098, 2186, 2278, 2382, 2472, 2560, 2688

Offset: 1

Alois P. Heinz, Jul 05 2024

nonn

Sum over all positive integers k, m with k*m <= n of phi(lcm(k,m)).

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Wikipedia, Iverson bracket

Partial sums of A061884.

Cf. A000010, A003990, A051173, A372866.

a:= proc(n) option remember; uses numtheory; `if`(n<1, 0,
      a(n-1)+add(phi(ilcm(d, n/d)), d=divisors(n)))
    end:
seq(a(n), n=1..66);

a(n) = Sum_{j=1..n} Sum_{d|j} phi(lcm(d,j/d)).

a(n) = Sum_{j=1..n} A061884(j).

A160401 Table read by antidiagonals: a(m,n) = the smallest composite multiple of both m and n.

Original entry on oeis.org

4, 4, 4, 6, 4, 6, 4, 6, 6, 4, 10, 4, 6, 4, 10, 6, 10, 12, 12, 10, 6, 14, 6, 15, 4, 15, 6, 14, 8, 14, 6, 20, 20, 6, 14, 8, 9, 8, 21, 12, 10, 12, 21, 8, 9, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 22, 10, 9, 8, 35, 6, 35, 8, 9, 10, 22, 12, 22, 30, 36, 40, 42, 42, 40, 36, 30, 22, 12, 26, 12, 33

Offset: 1

Leroy Quet, May 12 2009

a(p,1) = a(1,p) = a(p,p) = 2p, where p = any one prime. a(1,1) = 4. Otherwise, a(m,n) = lcm(m,n).

			Array begins:
   4,  4,  6,  4, 10,  6, ...
   4,  4,  6,  4, 10,  6, ...
   6,  6,  6, 12, 15,  6, ...
   4,  4, 12,  4, 20, 12, ...
  10, 10, 15, 20, 10, 30, ...
   6,  6,  6, 12, 30,  6, ...
  ...

Cf. A003990.

T(n, k) = {my(j = lcm(n, k), c = j); while (isprime(c) || (c==1), c += j); c;}
tabl(nn) = for (n=1, nn, for (k=1, nn, print1(T(n, k), ", ")); print); \\ Michel Marcus, Mar 13 2018

Extended by Ray Chandler, Jun 18 2009

cp's OEIS Frontend

A285724 Square array read by descending antidiagonals: If n > k, A(n,k) = T(lcm(n,k), gcd(n,k)), otherwise A(n,k) = T(gcd(n,k), lcm(n,k)), where T(n,k) is sequence A000027 considered as a two-dimensional table.

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A287958 Table read by antidiagonals: T(n, k) = least recursive multiple of n and k; n > 0 and k > 0.

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PARI

A339394 Sum over all partitions of n of the LCM of the number of parts and the number of distinct parts.

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A343033 Array T(n, k), n, k > 0, read by antidiagonals; a variant of lunar multiplication (A087062) based on prime exponents of numbers (see Comments section for precise definition).

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A082022 In the following square array a(i,j) = Least Common Multiple of i and j. Sequence contains the product of the terms of the n-th antidiagonal.

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Extensions

A264009 Table T(i,j) = nonnegative k at which lcm(i+k,j+k) reaches the minimum, read by antidiagonals (i>=1, j>=1).

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A305720 Square array T(n, k) read by antidiagonals, n > 0 and k > 0; for any prime number p, the p-adic valuation of T(n, k) is the product of the p-adic valuations of n and of k.

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A342905 Array read by antidiagonals: T(n,k) = product of all distinct primes dividing n*k (n>=1, k>=1).

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