

A321501


Numbers not of the form (x  y)(x^2  y^2) with x > y > 0; complement of A321499.


2



0, 1, 2, 4, 6, 8, 10, 12, 14, 18, 20, 22, 26, 28, 30, 34, 36, 38, 42, 44, 46, 50, 52, 54, 58, 60, 62, 66, 68, 70, 74, 76, 78, 82, 84, 86, 90, 92, 94, 98, 100, 102, 106, 108, 110, 114, 116, 118, 122, 124, 126, 130, 132, 134, 138, 140, 142, 146, 148, 150, 154, 156, 158, 162, 164, 166, 170, 172, 174, 178
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OFFSET

1,3


COMMENTS

Equivalently, numbers not of the form (x  y)^2*(x + y) or d^2*(2m + d), for (x, y) = (m+d, m). This shows that excluded are all squares d^2 > 0 times any number of the same parity and larger than d. In particular, for d=1, all odd numbers > 1, and for d=2, 4*(even numbers > 4) = 8*(odd numbers > 2). For larger d, no further (neither odd nor even) numbers are excluded.
So apart from 0, 1 and 8, this consists of even numbers not multiple of 8. All these numbers occur, since for larger (odd or even) d, no additional term is excluded.


LINKS

Table of n, a(n) for n=1..70.


FORMULA

Asymptotic density is 3/8.
a(n) = round((n2)*9/8)*2 for all n > 6.


EXAMPLE

a(1) = 0, a(2) = 1 and a(3) = 2 obviously can't be of the form (x  y)(x^2  y^2) with x > y > 0, which is necessarily greater than 1*3 = 3.
See A321499 for examples of the terms that are not in the sequence.


PROG

(PARI) is(n)={!n!fordiv(n, d, d^2*(d+2)>n && break; n%d^2&&next; bittest(n\d^2d, 0)return)} \\ Uses the initial definition. More efficient variant below:
(PARI) select( is_A321501(n)=!bittest(n, 0)&&(n%8n<9)n<3, [0..99]) \\ Defines the function is_A321501(). The select() command is an illustration and a check.
(PARI) A321501_list(M)={setunion([1], setminus([0..M\2]*2, [2..M\8]*8))} \\ Return all terms up to M; more efficient than to use select(..., [0..M]) as above.
(PARI) A321501(n)=if(n>6, (n2)*9\/8*2, n>3, n*24, n1)


CROSSREFS

See A321499 for the complement: numbers of the form (xy)(x^2y^2).
See A321491 for numbers of the form (x+y)(x^2+y^2).
Sequence in context: A240114 A076828 A276106 * A238264 A098807 A189562
Adjacent sequences: A321498 A321499 A321500 * A321502 A321503 A321504


KEYWORD

nonn,easy


AUTHOR

M. F. Hasler, Nov 22 2018


STATUS

approved



