A321501 Numbers not of the form (x - y)(x^2 - y^2) with x > y > 0; complement of A321499.

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

Offset: 1

Author

M. F. Hasler, Nov 22 2018

Keywords

• nonn
• easy

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.

Examples

			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.
		

Crossrefs

See A321499 for the complement: numbers of the form (x-y)(x^2-y^2).

See A321491 for numbers of the form (x+y)(x^2+y^2).

Programs

PARI

is(n)={!n||!fordiv(n,d, d^2*(d+2)>n && break; n%d^2&&next; bittest(n\d^2-d,0)||return)} \\ Uses the initial definition. More efficient variant below:

PARI

select( is_A321501(n)=!bittest(n,0)&&(n%8||n<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,(n-2)*9\/8*2,n>3,n*2-4,n-1)

Formula

Asymptotic density is 3/8.

a(n) = round((n-2)*9/8)*2 for all n > 6.