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A255483 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. 13
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

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

LINKS

Alois P. Heinz, Antidiagonals n = 0..125, flattened

C. Cobeli, A. Zaharescu, A game with divisors and absolute differences of exponents, Journal of Difference Equations and Applications, Vol. 20, #11, 2014.

C. Cobeli, A. Zaharescu, A game with divisors and absolute differences of exponents, arXiv:1411.1334 [math.NT], 2014.

Discussion of SeqFan-mailing list

FORMULA

T(n,1) = A123098(n), T(n,m+1) = A003961(T(n,m)), for all n >= 0, m >= 1. - M. F. Hasler, Sep 17 2016

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

From Antti Karttunen, Sep 18 2016: (Start)

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)).

T(n,k) = A007913(A066117(n+1,k)).

T(n,k) = A019565(A099884(n,k-1)) - After Hugo van der Sanden's observations on SeqFan-list.

(End)

EXAMPLE

The top left {1..10} x {0..5} corner of the array:

    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

MAPLE

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:

seq(seq(T(n, 1+d-n), n=0..d), d=0..10);  # Alois P. Heinz, Feb 28 2015

MATHEMATICA

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-Fran├žois Alcover, Mar 09 2015, after Alois P. Heinz *)

PROG

(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)

(define (A255483 n) (A255483bi (A002262 n) (+ 1 (A025581 n))))

;; Then use either a 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)))))

;; Or one based on M. F. Hasler's new recurrence:

(define (A255483bi row col) (if (= 1 col) (A123098 row) (A003961 (A255483bi row (- col 1)))))

;; Antti Karttunen, Sep 18 2016

CROSSREFS

First two columns = A123098, A276804.

Rows = A000040, A006094, A090076, A046302, ...

A kind of generalization of A036262.

Cf. A003961, A007913, A019565, A048675, A066117, A099884.

Transpose: A276578, terms sorted into ascending order: A276579.

Sequence in context: A067392 A066449 A276942 * A098012 A066117 A222311

Adjacent sequences:  A255480 A255481 A255482 * A255484 A255485 A255486

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane, Feb 28 2015

STATUS

approved

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Last modified September 17 10:52 EDT 2019. Contains 327129 sequences. (Running on oeis4.)