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A321501
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Numbers not of the form (x - y)(x^2 - y^2) with x > y > 0; complement of A321499.
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2
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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
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1,3
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COMMENTS
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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.
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LINKS
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FORMULA
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Asymptotic density is 3/8.
a(n) = round((n-2)*9/8)*2 for all n > 6.
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EXAMPLE
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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.
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PROG
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(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)
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CROSSREFS
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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).
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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