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%I #21 Aug 01 2019 10:54:17
%S 0,3,5,10,14,21,24,27,36,44,55,59,61,65,78,90,105,110,114,119,136,144,
%T 152,171,177,183,189,210,230,253,260,263,265,268,275,300,324,351,359,
%U 369,377,406,418,422,434,465,474,480,486,495,528,560,595,605,609,615,619,629
%N The image of N X N under f, where f(x,y) = 2*x*y*(x*y-1)-x+y.
%C f is the composite (P o D), where P(x,y)=((x+y)^2+3*x+y)/2 is the Cantor polynomial and D(x,y)=((x+1)*(y-1),(x-1)*(y+1)) is a divisor plot built so as to fit the first quadrant. This sequence can be viewed as an irregular table where the length of row n is A000005(n), the number of divisors of n.
%H Luc Rousseau, <a href="/A308805/a308805.svg">Illustration</a>
%F a(A006218(n)) = (n-1)*(2*n+1) = A014106(n-1), n >= 1.
%F a(A006218(n)+1) = n*(2*n+1) = A014105(n), n >= 0.
%F 24*A002415 is the subsequence made of all f(x,x), x >= 1.
%F n is prime iff (n-1)*(2*n-1) and (n-1)*(2*n+1) are consecutive terms in this sequence.
%e 1: 0
%e . .
%e 2: 3 . 5
%e . . . .
%e 3: 10 . . . 14
%e . . . . . .
%e 4: 21 . . 24 . . 27
%e . . . . . . . .
%e 5: 36 . . . . . . . 44
%e . . . . . . . . . .
%e 6: 55 . . . 59 . 61 . . . 65
%e . . . . . . . . . . . .
%e 7: 78 . . . . . . . . . . . 90
%e . . . . . . . . . . . . . .
%e 8: 105 . . . . 110 . . . 114 . . . . 119
%e . . . . . . . . . . . . . . . .
%e 9: 136 . . . . . . . 144 . . . . . . . 152
%e . . . . . . . . . . . . . . . . . .
%e (...)
%e This sequence is what remains when one removes the dots or "unoccupied integers" from the above schema, result of the superposition of a divisor plot on a triangle of numbered points.
%o (PARI)
%o f(x,y)=2*x*y*(x*y-1)-x+y
%o for(n=1,20,fordiv(n,d,print1(f(n/d,d),", ")))
%Y Cf. A000005, A006218, A014105, A014106, A002415.
%K nonn,tabf
%O 1,2
%A _Luc Rousseau_, Jun 25 2019
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