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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 (list; graph; refs; listen; history; text; internal format)
 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 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

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Last modified June 18 23:16 EDT 2021. Contains 345125 sequences. (Running on oeis4.)