A003990 -id:A003990 - cp's OEIS frontend

A089913 Table T(n,k) = lcm(n,k)/gcd(n,k) = n*k/gcd(n,k)^2 read by antidiagonals (n >= 1, k >= 1).

1, 2, 2, 3, 1, 3, 4, 6, 6, 4, 5, 2, 1, 2, 5, 6, 10, 12, 12, 10, 6, 7, 3, 15, 1, 15, 3, 7, 8, 14, 2, 20, 20, 2, 14, 8, 9, 4, 21, 6, 1, 6, 21, 4, 9, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 11, 5, 3, 2, 35, 1, 35, 2, 3, 5, 11, 12, 22, 30, 36, 40, 42, 42, 40, 36, 30, 22, 12, 13, 6, 33, 10, 45

Offset: 1

Marc LeBrun, Nov 14 2003

A multiplicative analog of absolute difference A049581. Exponents in prime factorization of T(n,k) are absolute differences of those of n and k. Commutative non-associative operator with identity 1. T(nx,kx)=T(n,k), T(n^x,k^x)=T(n,k)^x, etc.
The bivariate function log(T(., .)) is a distance (or metric) function. It is a weighted analog of A130836, in the sense that if e_i (resp. f_i) denotes the exponent of prime p_i in the factorization of m (resp. of n), then both log(T(m, n)) and A130836(m, n) are writable as Sum_{i} w_i * abs(e_i - f_i). For A130836, w_i = 1 for all i, whereas for log(T(., .)), w_i = log(p_i). - Luc Rousseau, Sep 17 2018
If the analog of absolute difference, as described in the first comment, is determined by factorization into distinct terms of A050376 instead of by prime factorization, the equivalent operation is defined by A059897 and is associative. The positive integers form a group under A059897. The two factorization methods give the same factorization for squarefree numbers (A005117), so that T(.,.) restricted to A005117 is associative. Thus the squarefree numbers likewise form a group under the operation defined by this sequence. - Peter Munn, Apr 04 2019

			T(6,10) = lcm(6,10)/gcd(6,10) = 30/2 = 15.
  1,  2,  3,  4,  5, ...
  2,  1,  6,  2, 10, ...
  3,  6,  1, 12, 15, ...
  4,  2, 12,  1, 20, ...
  5, 10, 15, 20,  1, ...
  ...

Ivan Neretin, Table of n, a(n) for n = 1..5050

Cf. A049581, A003990, A003991, A130836, A001222, A005117, A050376, A059897.

T:=Flat(List([1..13],n->List([1..n-1],k->Lcm(k,n-k)/Gcd(k,n-k)))); # Muniru A Asiru, Oct 24 2018

Flatten[Table[LCM[i, m - i]/GCD[i, m - i], {m, 15}, {i, m - 1}]] (* Ivan Neretin, Apr 27 2015 *)

A089913(n,k)=n*k/gcd(n,k)^2 \\ M. F. Hasler, Dec 06 2019

A130836(n, k) = A001222(T(n, k)). - Luc Rousseau, Sep 17 2018

A286101 Square array A(n,k) read by antidiagonals: A(n,k) = T(gcd(n,k), lcm(n,k)), where T(n,k) is sequence A000027 considered as a two-dimensional table.

Original entry on oeis.org

1, 2, 2, 4, 5, 4, 7, 16, 16, 7, 11, 12, 13, 12, 11, 16, 46, 67, 67, 46, 16, 22, 23, 106, 25, 106, 23, 22, 29, 92, 31, 191, 191, 31, 92, 29, 37, 38, 211, 80, 41, 80, 211, 38, 37, 46, 154, 277, 379, 436, 436, 379, 277, 154, 46, 56, 57, 58, 59, 596, 61, 596, 59, 58, 57, 56, 67, 232, 436, 631, 781, 862, 862, 781, 631, 436, 232, 67, 79, 80, 529, 212, 991, 302, 85, 302, 991, 212, 529, 80, 79

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
   2,   5,  16,  12,   46,   23,   92,   38,  154,   57,  232,   80
   4,  16,  13,  67,  106,   31,  211,  277,   58,  436,  529,   94
   7,  12,  67,  25,  191,   80,  379,   59,  631,  212,  947,  109
  11,  46, 106, 191,   41,  436,  596,  781,  991,   96, 1486, 1771
  16,  23,  31,  80,  436,   61,  862,  302,  193,  467, 2146,  142
  22,  92, 211, 379,  596,  862,   85, 1541, 1954, 2416, 2927, 3487
  29,  38, 277,  59,  781,  302, 1541,  113, 2557,  822, 3829,  355
  37, 154,  58, 631,  991,  193, 1954, 2557,  145, 4006, 4852,  706
  46,  57, 436, 212,   96,  467, 2416,  822, 4006,  181, 5996, 1832
  56, 232, 529, 947, 1486, 2146, 2927, 3829, 4852, 5996,  221, 8647
  67,  80,  94, 109, 1771,  142, 3487,  355,  706, 1832, 8647,  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 and column 1), A001844 (main diagonal).

Cf. A000027, A003989, A003990, A003991, A038722, A285722, A285732, A286098, A286099, A286102, A285724.

(define (A286101 n) (A286101bi (A002260 n) (A004736 n)))
(define (A286101bi row col) (A000027bi (gcd row col) (lcm row col)))
(define (A000027bi row col) (* (/ 1 2) (+ (expt (+ row col) 2) (- row) (- (* 3 col)) 2)))

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.

A(n,k) = A(k,n), or equivalently, a(A038722(n)) = a(n). [Array is symmetric.]

A006580 a(n) = Sum_{k=1..n-1} lcm(k,n-k).

Original entry on oeis.org

0, 0, 1, 4, 8, 20, 21, 56, 60, 96, 105, 220, 152, 364, 301, 360, 464, 816, 549, 1140, 760, 1036, 1221, 2024, 1196, 2200, 2041, 2484, 2184, 4060, 2205, 4960, 3664, 4224, 4641, 5180, 4008, 8436, 6517, 7072, 5980, 11480, 6321, 13244, 8888, 9540, 11661, 17296

Offset: 0

N. J. A. Sloane

nonn

Marc LeBrun, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1000 terms from Reinhard Zumkeller)
Marc Le Brun, Email to N. J. A. Sloane, Jul 1991.

Antidiagonal sums of array A003990.
Cf. A209295.
Cf. A000010, A023900, A057660, A130054.

a006580 n = a006580_list !! (n-1)
a006580_list = map sum a003990_tabl
-- Reinhard Zumkeller, Aug 05 2012

a:= n-> add(ilcm(j, n-j), j=0..n):
seq(a(n), n=0..70);  # Alois P. Heinz, Aug 25 2019

Table[ Sum[ LCM[ k, n-k ], {k, 1, n-1} ], {n, 2, 50} ] (* Olivier Gérard, Aug 15 1997 *)
f1[p_, e_] := (p^(2*e + 1) + 1)/(p + 1); f2[p_, e_] := 1 - (p - 1)*e; a[n_] := (Times @@ f1 @@@ (fct = FactorInteger[n]) - Times @@ f2 @@@ fct)*n/6; a[0] = 0; Array[a, 100, 0] (* Amiram Eldar, Apr 28 2023 *)

a(n) = sum(k=1, n-1, lcm(k, n-k)); \\ Michel Marcus, Aug 11 2017

For n > 0, a(n) = (n/6)*Sum_{d|n} (d*phi(d) - A023900(d)). - Sebastian Karlsson, Oct 02 2021

a(n) = (n/6) * (A057660(n) - A130054(n)), for n > 0. - Amiram Eldar, Apr 28 2023

More terms from Olivier Gérard, Aug 15 1997

A075175 Prime factorization of n encoded by interleaving successive prime exponents in unary to bit-positions given by columns of A001477.

Original entry on oeis.org

0, 1, 2, 5, 8, 3, 64, 37, 18, 9, 1024, 7, 32768, 65, 10, 549, 2097152, 19, 268435456, 13, 66, 1025, 68719476736, 39, 136, 32769, 274, 69, 35184372088832, 11, 36028797018963968, 16933, 1026, 2097153, 72, 23, 73786976294838206464

Offset: 1

Antti Karttunen, Sep 13 2002

nonn

Here we store the exponent e_i of p_i (p1=2, p2=3, p3=5, ...) in the factorization of n to the bit positions given by the column i-1 of A001477 viewed as a table (the exponent of 2 is thus stored to bit positions 0, 2, 5, 9, 14, 20, ..., exponent of 3 to 1, 4, 8, 13, 19, ..., exponent of 5 to 3, 7, 12, 18, 25, ...) using unary system, i.e. we actually store 2^(e_i) - 1 in binary.

This injective mapping from N to N offers an example of the proof given in Cameron's book that any distributive lattice can be represented as a sublattice of the power-set lattice P(X) of some set X. With this we can implement GCD (A003989) with bitwise AND (A004198) and LCM (A003990) with bitwise OR (A003986). Also, to test whether x divides y, it is enough to check that ((a(x) OR a(y)) XOR a(y)) = A003987(A003986(a(x),a(y)),a(y)) is zero.

			a(24) = 39 because 24 = 2^3 * 3^1 so we add the binary words 100101 and 10 to get 100111 in binary = 39 in decimal and a(25) = 136 because 25 = 5^2 so we form a binary word 10001000 = 136 in decimal.

P. J. Cameron, Combinatorics: Topics, Techniques, Algorithms, Cambridge University Press, 1998, page 191. (12.3. Distributive lattices)

Variant: A075173. Inverse: A075176.

A003989(x, y) = A075176(A004198(a(x), a(y))), A003990(x, y) = A075176(A003986(a(x), a(y))).

A188649 Least common multiple of reversals of divisors of n in decimal representation.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 84, 31, 574, 255, 488, 71, 162, 91, 20, 84, 22, 32, 168, 260, 62, 72, 1148, 92, 510, 13, 11224, 33, 6106, 1855, 2268, 73, 15106, 93, 40, 14, 6888, 34, 44, 4590, 64, 74, 10248, 658, 260, 1065, 3100, 35, 3240, 55, 149240, 6825, 7820, 95, 7140, 16, 26, 252, 11224, 8680, 66, 76, 12212, 96, 152110, 17, 4536, 37, 6862, 251940, 2024204, 77

Offset: 1

Reinhard Zumkeller, Apr 11 2011

			Divisors(42) = {1,2,3,6,7,14,21,42}, therefore a(42) = lcm(1,2,3,6,7,41,12,24) = 6888.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to lcm's

Cf. A003990 (LCM), A027750 (divisors), A004086 (reversal), A188650.

Cf. A000040, A004087, A002385.

a188649 n = foldl lcm 1 $ map a004086 $ filter ((== 0) . mod n) [1..n]

a(n) = lcm(apply(x->fromdigits(Vecrev(digits(x))), divisors(n))); \\ Michel Marcus, Mar 13 2018

from math import lcm
from sympy import divisors
def a(n): return lcm(*(int(str(d)[::-1]) for d in divisors(n)))
print([a(n) for n in range(1, 78)]) # Michael S. Branicky, Aug 14 2022

a(A000040(n)) = A004087(n);

a(A002385(n)) = A002385(n), see A188650 for all fixed points.

A286102 Square array A(n,k) read by antidiagonals: A(n,k) = T(lcm(n,k), gcd(n,k)), where T(n,k) is sequence A000027 considered as a two-dimensional table.

Original entry on oeis.org

1, 3, 3, 6, 5, 6, 10, 21, 21, 10, 15, 14, 13, 14, 15, 21, 55, 78, 78, 55, 21, 28, 27, 120, 25, 120, 27, 28, 36, 105, 34, 210, 210, 34, 105, 36, 45, 44, 231, 90, 41, 90, 231, 44, 45, 55, 171, 300, 406, 465, 465, 406, 300, 171, 55, 66, 65, 64, 63, 630, 61, 630, 63, 64, 65, 66, 78, 253, 465, 666, 820, 903, 903, 820, 666, 465, 253, 78, 91, 90, 561, 230, 1035, 324, 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,   3,   6,  10,   15,   21,   28,   36,   45,   55,   66,   78
   3,   5,  21,  14,   55,   27,  105,   44,  171,   65,  253,   90
   6,  21,  13,  78,  120,   34,  231,  300,   64,  465,  561,  103
  10,  14,  78,  25,  210,   90,  406,   63,  666,  230,  990,  117
  15,  55, 120, 210,   41,  465,  630,  820, 1035,  101, 1540, 1830
  21,  27,  34,  90,  465,   61,  903,  324,  208,  495, 2211,  148
  28, 105, 231, 406,  630,  903,   85, 1596, 2016, 2485, 3003, 3570
  36,  44, 300,  63,  820,  324, 1596,  113, 2628,  860, 3916,  375
  45, 171,  64, 666, 1035,  208, 2016, 2628,  145, 4095, 4950,  739
  55,  65, 465, 230,  101,  495, 2485,  860, 4095,  181, 6105, 1890
  66, 253, 561, 990, 1540, 2211, 3003, 3916, 4950, 6105,  221, 8778
  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. A000027, A003989, A003990, A003991, A038722, A285722, A285732, A286098, A286099, A286101, A285724.

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

(define (A286102 n) (A286102bi (A002260 n) (A004736 n)))
(define (A286102bi row col) (A000027bi (lcm row col) (gcd row col)))
(define (A000027bi row col) (* (/ 1 2) (+ (expt (+ row col) 2) (- row) (- (* 3 col)) 2)))

A(n,k) = T(lcm(n,k), gcd(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.

A(n,k) = A(k,n), or equivalently, a(A038722(n)) = a(n). [Array is symmetric.]

A091256 Table of lcm(x,y) computed for polynomials over GF(2), where (x,y) runs as (1,1),(1,2),(2,1),(1,3),(2,2),(3,1),...

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, 5, 4, 5, 6, 7, 8, 14, 6, 20, 20, 6, 14, 8, 9, 8, 9, 12, 5, 12, 9, 8, 9, 10, 18, 24, 28, 10, 10, 28, 24, 18, 10, 11, 10, 9, 8, 27, 6, 27, 8, 9, 10, 11, 12, 22, 10, 36, 40, 18, 18, 40, 36, 10, 22, 12, 13, 12, 29

Offset: 1

Antti Karttunen, Jan 03 2004

Analogous to A003990.

Cf. A091255, A091257.

A157754 a(1) = 0, a(n) = lcm(A051904(n), A051903(n)) for n >= 2.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 5, 1, 2, 2, 2

Offset: 1

Jaroslav Krizek, Mar 05 2009

nonn

a(n) for n >= 2 equals LCM of minimum and maximum exponents in the prime factorization of n.

a(n) for n >= 2 deviates from A072411, first different term is a(360), a(360) = 3, A072411(360) = 6.

			For n = 12 = 2^2 * 3^1 we have a(12) = lcm(2,1) = 2.
For n = 144 = 2^4 * 3^2 we have a(144) = lcm(4,2) = 4.

Cf. A000040, A003990, A006881, A033150, A120944, A000961, A000027, A072411, A051904, A051903, A158378.

Table[LCM @@ {Min@ #, Max@ #} - Boole[n == 1] &@ FactorInteger[n][[All, -1]], {n, 100}] (* Michael De Vlieger, Jul 12 2017 *)

a(n) = if(n == 1, 0, my(e = factor(n)[,2]); lcm(vecmin(e), vecmax(e))); \\ Amiram Eldar, Sep 11 2024

a(1) = 0, a(p) = 1, a(pq) = 1, a(pq...z) = 1, a(p^k) = k, for p = primes (A000040), pq = product of two distinct primes (A006881), pq...z = product of k (k > 2) distinct primes p, q, ..., z (A120944), p^k = prime powers (A000961(n) for n > 1) k = natural numbers (A000027).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A033150. - Amiram Eldar, Sep 11 2024

A351962 Square array A(n,k) = A156552(lcm(A005940(1+n), A005940(1+k))), read by antidiagonals.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 3, 5, 5, 3, 4, 3, 2, 3, 4, 5, 9, 11, 11, 9, 5, 6, 5, 10, 3, 10, 5, 6, 7, 13, 5, 19, 19, 5, 13, 7, 8, 7, 6, 11, 4, 11, 6, 7, 8, 9, 17, 23, 27, 21, 21, 27, 23, 17, 9, 10, 9, 18, 7, 22, 5, 22, 7, 18, 9, 10, 11, 21, 21, 35, 39, 13, 13, 39, 35, 21, 21, 11, 12, 11, 10, 19, 20, 23, 6, 23, 20, 19, 10, 11, 12

Offset: 0

Antti Karttunen, Feb 26 2022

The indices run as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), etc. The array is symmetric.

			The top left corner of the array:
      n=0   1   2   3   4   5    6    7    8   9   10   11   12
-----|-----------------------------------------------------------
k= 0 |  0,  1,  2,  3,  4,  5,   6,   7,   8,  9,  10,  11,  12,
   1 |  1,  1,  5,  3,  9,  5,  13,   7,  17,  9,  21,  11,  25,
   2 |  2,  5,  2, 11, 10,  5,   6,  23,  18, 21,  10,  11,  26,
   3 |  3,  3, 11,  3, 19, 11,  27,   7,  35, 19,  43,  11,  51,
   4 |  4,  9, 10, 19,  4, 21,  22,  39,  20,  9,  10,  43,  12,
   5 |  5,  5,  5, 11, 21,  5,  13,  23,  37, 21,  21,  11,  53,
   6 |  6, 13,  6, 27, 22, 13,   6,  55,  38, 45,  22,  27,  54,
   7 |  7,  7, 23,  7, 39, 23,  55,   7,  71, 39,  87,  23, 103,
   8 |  8, 17, 18, 35, 20, 37,  38,  71,   8, 41,  42,  75,  44,
   9 |  9,  9, 21, 19,  9, 21,  45,  39,  41,  9,  21,  43,  25,
  10 | 10, 21, 10, 43, 10, 21,  22,  87,  42, 21,  10,  43,  26,
  11 | 11, 11, 11, 11, 43, 11,  27,  23,  75, 43,  43,  11, 107,
  12 | 12, 25, 26, 51, 12, 53,  54, 103,  44, 25,  26, 107,  12,
  13 | 13, 13, 13, 27, 45, 13,  13,  55,  77, 45,  45,  27, 109,
  14 | 14, 29, 14, 59, 46, 29,  14, 119,  78, 93,  46,  59, 110,
  15 | 15, 15, 47, 15, 79, 47, 111,  15, 143, 79, 175,  47, 207,
  16 | 16, 33, 34, 67, 36, 69,  70, 135,  40, 73,  74, 139,  76,

Cf. A003990, A005940, A156552.
Cf. A001477 (main diagonal).
Cf. also A341520, A351960, A351961.

up_to = 104;
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A351962sq(n,k) = A156552(lcm(A005940(1+n),A005940(1+k)));
A351962list(up_to) = { my(v = vector(1+up_to), i=0); for(a=0,oo, for(col=0,a, i++; if(i > #v, return(v)); v[i] = A351962sq(col,(a-(col))))); (v); };
v351962 = A351962list(up_to);
A351962(n) = v351962[1+n];

A257522 Table T(i,j) = denominator of (1/i + 1/j) = ij/gcd(ij,i+j) read by antidiagonals (i >= 1, j >= 1).

Original entry on oeis.org

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

Offset: 1

Ivan Neretin, Apr 27 2015

a(n) is a divisor of A003990(n) and a multiple of A089913(n).

Ivan Neretin, Table of n, a(n) for n = 1..5050

Cf. A003991, A003990, A089913.

Flatten[Table[Denominator[1/i + 1/(m - i)], {m, 15}, {i, m - 1}]]

cp's OEIS Frontend

A089913 Table T(n,k) = lcm(n,k)/gcd(n,k) = n*k/gcd(n,k)^2 read by antidiagonals (n >= 1, k >= 1).

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A286101 Square array A(n,k) read by antidiagonals: 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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A006580 a(n) = Sum_{k=1..n-1} lcm(k,n-k).

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A075175 Prime factorization of n encoded by interleaving successive prime exponents in unary to bit-positions given by columns of A001477.

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A188649 Least common multiple of reversals of divisors of n in decimal representation.

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A286102 Square array A(n,k) read by antidiagonals: A(n,k) = T(lcm(n,k), gcd(n,k)), where T(n,k) is sequence A000027 considered as a two-dimensional table.

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A091256 Table of lcm(x,y) computed for polynomials over GF(2), where (x,y) runs as (1,1),(1,2),(2,1),(1,3),(2,2),(3,1),...

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A157754 a(1) = 0, a(n) = lcm(A051904(n), A051903(n)) for n >= 2.

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A257522 Table T(i,j) = denominator of (1/i + 1/j) = ij/gcd(ij,i+j) read by antidiagonals (i >= 1, j >= 1).