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%I #28 Dec 07 2016 23:33:50
%S 1,2,2,3,6,3,6,2,12,4,5,30,15,20,5,15,3,10,6,30,6,105,7,21,210,35,42,
%T 7,70,6,42,2,420,12,56,8,1,70,105,10,5,84,63,72,9,5,5,14,30,3,15,1260,
%U 20,90,10,33,165,33,462,385,1155,77,1980,99,110,11,55,15,11,3,154,10,462,6,330,30,132,12,65,143,2145,195,65,10010,1001,78
%N Array read by antidiagonals: first row is 1, 2, 3, 4, ...; for subsequent rows, write i*j/gcd(i,j)^2 under ...i.j... in previous row.
%H Ivan Neretin, <a href="/A222310/b222310.txt">Table of n, a(n) for n = 1..8001</a>
%H Cristian Cobeli, Mihai Prunescu, Alexandru Zaharescu, <a href="http://arxiv.org/abs/1511.04315">A growth model based on the arithmetic Z-game</a>, arXiv:1511.04315 [math.NT], 2015.
%H C. Cobeli and A. Zaharescu, <a href="http://rms.unibuc.ro/bulletin/pdf/56-1/PromenadePascalPart1.pdf">Promenade around Pascal Triangle-Number Motives</a>, Bull. Math. Soc. Sci. Math. Roumanie, Tome 56(104) No. 1, 2013, 73-98.
%e Array begins:
%e 1...2...3.....4......5......6.....7.....8.....9.....10
%e ..2...6....12....20.....30....42.....56...72.....90
%e ....3...2....15......6.....35....12....63....20
%e ......6....30....10....210...420.....84..1260
%e ........5.....3.....21......2.....5....15
%e ...........15.....7.....42....10......3
%e .............105.....6.....105...30
%e ........
%p # To get first M rows of the array (s0 is A222311):
%p g:=(i,j)->i*j/gcd(i,j)^2;
%p M:=50;
%p s0:=[1]:
%p s1:=[seq(n,n=1..M)]:
%p for i1 from 1 to M-1 do
%p lprint(s1);
%p s2:=[seq(g(s1[i],s1[i+1]),i=1..nops(s1)-1)];
%p s0:=[op(s0),s2[1]];
%p s1:=[seq(s2[i],i=1..nops(s2))];
%p od:
%p # To produce A222310 (i.e., to read the array by antidiagonals):
%p g:=(i,j)->i*j/gcd(i,j)^2;
%p M:=15;
%p b1:=Array(1..M);
%p s0:=[1]:
%p s1:=[seq(n,n=1..M)]:
%p b1[1]:=s1;
%p for i1 from 1 to M-1 do
%p #lprint(s1);
%p s2:=[seq(g(s1[i],s1[i+1]),i=1..nops(s1)-1)];
%p b1[i1+1]:=s2;
%p s0:=[op(s0),s2[1]];
%p s1:=[seq(s2[i],i=1..nops(s2))];
%p od:
%p #[seq(s0[i],i=1..nops(s0))]; (that gives A222311)
%p lis:=[]:
%p for i from 1 to M do for j from 1 to i do
%p lis:=[op(lis),b1[i-j+1][j]];
%p od: od:
%p [seq(lis[k],k=1..nops(lis))];
%t a = r = {1}; Do[a = Join[a, Reverse[r = FoldList[#1*#2/GCD[#1, #2]^2 &, n, r]]], {n, 2, 13}]; a (* _Ivan Neretin_, May 14 2015 *)
%Y Cf. A036262. Leading diagonal is A222311 (cf. A222313).
%Y Similar array with primes in the starting row is A255483.
%K nonn,tabl
%O 1,2
%A _N. J. A. Sloane_, Feb 16 2013
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