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A308805
The image of N X N under f, where f(x,y) = 2*x*y*(x*y-1)-x+y.
2
0, 3, 5, 10, 14, 21, 24, 27, 36, 44, 55, 59, 61, 65, 78, 90, 105, 110, 114, 119, 136, 144, 152, 171, 177, 183, 189, 210, 230, 253, 260, 263, 265, 268, 275, 300, 324, 351, 359, 369, 377, 406, 418, 422, 434, 465, 474, 480, 486, 495, 528, 560, 595, 605, 609, 615, 619, 629
OFFSET
1,2
COMMENTS
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.
FORMULA
a(A006218(n)) = (n-1)*(2*n+1) = A014106(n-1), n >= 1.
a(A006218(n)+1) = n*(2*n+1) = A014105(n), n >= 0.
24*A002415 is the subsequence made of all f(x,x), x >= 1.
n is prime iff (n-1)*(2*n-1) and (n-1)*(2*n+1) are consecutive terms in this sequence.
EXAMPLE
1: 0
. .
2: 3 . 5
. . . .
3: 10 . . . 14
. . . . . .
4: 21 . . 24 . . 27
. . . . . . . .
5: 36 . . . . . . . 44
. . . . . . . . . .
6: 55 . . . 59 . 61 . . . 65
. . . . . . . . . . . .
7: 78 . . . . . . . . . . . 90
. . . . . . . . . . . . . .
8: 105 . . . . 110 . . . 114 . . . . 119
. . . . . . . . . . . . . . . .
9: 136 . . . . . . . 144 . . . . . . . 152
. . . . . . . . . . . . . . . . . .
(...)
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.
PROG
(PARI)
f(x, y)=2*x*y*(x*y-1)-x+y
for(n=1, 20, fordiv(n, d, print1(f(n/d, d), ", ")))
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Luc Rousseau, Jun 25 2019
STATUS
approved