OFFSET
1,1
COMMENTS
Equivalently, numbers of the form (x - y)^2*(x + y) or d^2*(2m + d), for (x, y) = (m+d, m). This shows that this consists of all squares d^2 > 0 times all numbers of the same parity and larger than d. In particular, for d=1, all odd numbers > 1, and for d=2, 4*(even numbers > 2) = 8*(any number > 1). Larger d can't yield additional terms, neither odd nor even: The sequence consists exactly of all odd numbers > 2 and multiples of 8 larger than 8.
LINKS
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).
FORMULA
Asymptotic density is 5/8. Complement is A321501.
a(5k-2) = 8k for all k > 1, a(n) = floor((n+2)*4/5)*2 + 1 for all other n > 3.
a(n + 5) = a(n) + 8 for n > 3. - David A. Corneth, Nov 23 2018
O.g.f. 3*x+5*x^2+7*x^3 -x^4*(-9-2*x-2*x^2-2*x^3-x^4+8*x^5) / ( (x^4+x^3+x^2+x+1) *(x-1)^2 ). - R. J. Mathar, Nov 29 2018
EXAMPLE
a(1) = 3 = 1*3 = (2 - 1)*(2^2 - 1^2). Similarly any larger odd number 2k+1 = (k+1 - k)((k+1)^2 - k^2) is in this sequence.
a(8) = 16 = 2*8 = (3 - 1)*(3^2 - 1^2). Similarly, any larger multiple of 8, 8*(1 + k) = 2*(4k + 4) = (k+2 - k)((k+2)^2 - k^2) is in this sequence.
PROG
(PARI) is(n)={n&&fordiv(n, d, d^2*(d+2)>n && break; n%d^2&&next; bittest(n\d^2-d, 0)||return(1))} \\ This uses the definition. More efficient variant below.
(PARI) select( is_A321499(n)=if(bittest(n, 0), n>1, n%8, 0, n>8), [0..99]) \\ Defines the function is_A321499(). The select() command is just an illustration and check.
(PARI) A321499_list(M)=setunion(vector(M\2-1, k, 2*k+1), [2..M\8]*8) \\ list all terms up to M; more efficient than select() above.
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
M. F. Hasler, Nov 22 2018
STATUS
approved