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A255483
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Infinite square array read by antidiagonals downwards: T(0,m) = prime(m), m >= 1; for n >= 1, T(n,m) = T(n-1,m)*T(n-1,m+1)/gcd(T(n-1,m), T(n-1,m+1))^2, m >= 1.
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14
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2, 3, 6, 5, 15, 10, 7, 35, 21, 210, 11, 77, 55, 1155, 22, 13, 143, 91, 5005, 39, 858, 17, 221, 187, 17017, 85, 3315, 1870, 19, 323, 247, 46189, 133, 11305, 5187, 9699690, 23, 437, 391, 96577, 253, 33649, 21505, 111546435, 46
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OFFSET
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0,1
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COMMENTS
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The first column of the array is given by A123098; subsequent columns are obtained by applying the function A003961, i.e., replacing each prime factor by the next larger prime. - M. F. Hasler, Sep 17 2016
With respect to the ring defined by A329329 and A059897, the first row gives powers of 3, the first column gives powers of 6, both in order of increasing exponent, and the body of the table gives their products. A329049 is the equivalent table in which the first column gives powers of 4.
A099884 is the equivalent table for the ring defined by A048720 and A003987. That ring is an image of the polynomial ring GF(2)[x] using a standard representation of the polynomials as integers. A329329 describes a comparable mapping to integers from the related polynomial ring GF(2)[x,y].
Using these mappings, the tables here and in A099884 are matching images: the first row represents powers of x, the first column represents powers of (x+1) and the body of the table gives their products.
Hugo van der Sanden's formula (see formula section) indicates that A019565 provides a mapping from A099884. In the wider terms described above, A019565 is an injective homomorphism between images of the 2 polynomial rings, and maps the image of each GF(2)[x] polynomial to the image of the equivalent GF(2)[x,y] polynomial.
(End)
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LINKS
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FORMULA
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T(n,m) = Prod_{k=0..n} prime(k+m)^(!(n-k & k)) where !x is 1 if x=0 and 0 else, and & is binary AND. - M. F. Hasler, Sep 18 2016
For n >= 1, m >= 1, T(n,m) = lcm(T(n-1,m),T(n-1,m+1)) / gcd(T(n-1,m),T(n-1,m+1)).
(End)
T(0,1) = 2, and for n >= 0, k >= 1, T(n+1,k) = A329329(T(n,k), 6), T(n,k+1) = A329329(T(n,k), 3).
(End)
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EXAMPLE
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The top left corner of the array, row index 0..5, column index 1..10:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29
6, 15, 35, 77, 143, 221, 323, 437, 667, 899
10, 21, 55, 91, 187, 247, 391, 551, 713, 1073
210, 1155, 5005, 17017, 46189, 96577, 215441, 392863, 765049, 1363783
22, 39, 85, 133, 253, 377, 527, 703, 943, 1247
858, 3315, 11305, 33649, 95381, 198679, 370481, 662929, 1175921, 1816879
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MAPLE
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T:= proc(n, m) option remember; `if`(n=0, ithprime(m),
T(n-1, m)*T(n-1, m+1)/igcd(T(n-1, m), T(n-1, m+1))^2)
end:
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MATHEMATICA
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T[n_, m_] := T[n, m] = If[n == 0, Prime[m], T[n-1, m]*T[n-1, m+1]/GCD[T[n-1, m], T[n-1, m+1]]^2]; Table[Table[T[n, 1+d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Mar 09 2015, after Alois P. Heinz *)
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PROG
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(PARI) T=matrix(N=15, N); for(j=1, N, T[1, j]=prime(j)); (f(x, y)=x*y/gcd(x, y)^2); for(k=1, N-1, for(j=1, N-k, T[k+1, j]=f(T[k, j], T[k, j+1]))); A255483=concat(vector(N, i, vector(i, j, T[j, 1+i-j]))) \\ M. F. Hasler, Sep 17 2016
(PARI) A255483(n, k)=prod(j=0, n, if(bitand(n-j, j), 1, prime(j+k))) \\ M. F. Hasler, Sep 18 2016
(Scheme)
;; Then use either an almost standalone version (requiring only A000040):
(define (A255483bi row col) (if (zero? row) (A000040 col) (let ((a (A255483bi (- row 1) col)) (b (A255483bi (- row 1) (+ col 1)))) (/ (lcm a b) (gcd a b)))))
(define (A255483bi row col) (if (= 1 col) (A123098 row) (A003961 (A255483bi row (- col 1)))))
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CROSSREFS
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A kind of generalization of A036262.
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KEYWORD
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AUTHOR
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STATUS
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approved
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