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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.

%I A255483 #68 Dec 23 2024 14:53:44
%S A255483 2,3,6,5,15,10,7,35,21,210,11,77,55,1155,22,13,143,91,5005,39,858,17,
%T A255483 221,187,17017,85,3315,1870,19,323,247,46189,133,11305,5187,9699690,
%U A255483 23,437,391,96577,253,33649,21505,111546435,46
%N 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.
%C A255483 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
%C A255483 Interpretation with respect to A329329 from _Peter Munn_, Feb 08 2020: (Start)
%C A255483 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.
%C A255483 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].
%C A255483 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.
%C A255483 _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.
%C A255483 (End)
%H A255483 Alois P. Heinz, <a href="/A255483/b255483.txt">Antidiagonals n = 0..125, flattened</a>
%H A255483 C. Cobeli, A. Zaharescu, <a href="http://dx.doi.org/10.1080/10236198.2014.940337">A game with divisors and absolute differences of exponents</a>, Journal of Difference Equations and Applications, Vol. 20, #11, 2014.
%H A255483 C. Cobeli, A. Zaharescu, <a href="http://arxiv.org/abs/1411.1334">A game with divisors and absolute differences of exponents</a>, arXiv:1411.1334 [math.NT], 2014.
%H A255483 <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2016-September/016788.html">Discussion of SeqFan-mailing list</a>
%H A255483 <a href="/index/Ge#Gilbreath">Index entries for sequences related to Gilbreath conjecture and transform</a>
%F A255483 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
%F A255483 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
%F A255483 From _Antti Karttunen_, Sep 18 2016: (Start)
%F A255483 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)).
%F A255483 T(n,k) = A007913(A066117(n+1,k)).
%F A255483 T(n,k) = A019565(A099884(n,k-1)) [After _Hugo van der Sanden_'s observations on SeqFan-list].
%F A255483 (End)
%F A255483 From _Peter Munn_, Jan 08 2020: (Start)
%F A255483 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).
%F A255483 T(n,k) = A329329(T(n,1), T(0,k)).
%F A255483 (End)
%e A255483 The top left corner of the array, row index 0..5, column index 1..10:
%e A255483     2,    3,     5,     7,    11,     13,     17,     19,      23,      29
%e A255483     6,   15,    35,    77,   143,    221,    323,    437,     667,     899
%e A255483    10,   21,    55,    91,   187,    247,    391,    551,     713,    1073
%e A255483   210, 1155,  5005, 17017, 46189,  96577, 215441, 392863,  765049, 1363783
%e A255483    22,   39,    85,   133,   253,    377,    527,    703,     943,    1247
%e A255483   858, 3315, 11305, 33649, 95381, 198679, 370481, 662929, 1175921, 1816879
%p A255483 T:= proc(n, m) option remember; `if`(n=0, ithprime(m),
%p A255483       T(n-1, m)*T(n-1, m+1)/igcd(T(n-1, m), T(n-1, m+1))^2)
%p A255483     end:
%p A255483 seq(seq(T(n, 1+d-n), n=0..d), d=0..10);  # _Alois P. Heinz_, Feb 28 2015
%t A255483 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_ *)
%o A255483 (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
%o A255483 (PARI) A255483(n,k)=prod(j=0,n,if(bitand(n-j,j),1,prime(j+k))) \\ _M. F. Hasler_, Sep 18 2016
%o A255483 (Scheme)
%o A255483 (define (A255483 n) (A255483bi (A002262 n) (+ 1 (A025581 n))))
%o A255483 ;; Then use either an almost standalone version (requiring only A000040):
%o A255483 (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)))))
%o A255483 ;; Or one based on _M. F. Hasler_'s new recurrence:
%o A255483 (define (A255483bi row col) (if (= 1 col) (A123098 row) (A003961 (A255483bi row (- col 1)))))
%o A255483 ;; _Antti Karttunen_, Sep 18 2016
%Y A255483 First two columns = A123098, A276804.
%Y A255483 Rows = A000040, A006094, A090076, A046302, ...
%Y A255483 A kind of generalization of A036262.
%Y A255483 Cf. A003961, A007913, A019565, A048675, A066117, A099884.
%Y A255483 Transpose: A276578, terms sorted into ascending order: A276579.
%Y A255483 A003987, A048720, A059897, A329049 relate to the A329329 polynomial ring interpretation.
%K A255483 nonn,tabl
%O A255483 0,1
%A A255483 _N. J. A. Sloane_, Feb 28 2015