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


%S 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,

%T 60,62,66,68,70,74,76,78,82,84,86,90,92,94,98,100,102,106,108,110,114,

%U 116,118,122,124,126,130,132,134,138,140,142,146,148,150,154,156,158,162,164,166,170,172,174,178

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

%C 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.

%C 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.

%F Asymptotic density is 3/8.

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

%e 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.

%e See A321499 for examples of the terms that are not in the sequence.

%o (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:

%o (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.

%o (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.

%o (PARI) A321501(n)=if(n>6,(n-2)*9\/8*2,n>3,n*2-4,n-1)

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

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

%K nonn,easy

%O 1,3

%A _M. F. Hasler_, Nov 22 2018

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Last modified April 6 07:02 EDT 2020. Contains 333267 sequences. (Running on oeis4.)