A308805 The image of N X N under f, where f(x,y) = 2*x*y*(x*y-1)-x+y.

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.

Links

Table of n, a(n) for n=1..58.

Luc Rousseau, Illustration

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

Cf. A000005, A006218, A014105, A014106, A002415.

Sequence in context: A310018 A048214 A195094 * A001841 A266793 A176222

Adjacent sequences: A308802 A308803 A308804 * A308806 A308807 A308808

Keyword

nonn,tabf

Author

Luc Rousseau, Jun 25 2019

Status

approved