cp's OEIS Frontend

A321499 Numbers of the form (x - y)(x^2 - y^2) with x > y > 0.

%I A321499 #24 Feb 26 2025 16:45:28
%S A321499 3,5,7,9,11,13,15,16,17,19,21,23,24,25,27,29,31,32,33,35,37,39,40,41,
%T A321499 43,45,47,48,49,51,53,55,56,57,59,61,63,64,65,67,69,71,72,73,75,77,79,
%U A321499 80,81,83,85,87,88,89,91,93,95,96,97,99,101,103,104,105,107,109,111,112,113,115
%N A321499 Numbers of the form (x - y)(x^2 - y^2) with x > y > 0.
%C A321499 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.
%H A321499 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,1,-1).
%F A321499 Asymptotic density is 5/8. Complement is A321501.
%F A321499 a(5k-2) = 8k for all k > 1, a(n) = floor((n+2)*4/5)*2 + 1 for all other n > 3.
%F A321499 a(n + 5) = a(n) + 8 for n > 3. - _David A. Corneth_, Nov 23 2018
%F A321499 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
%e A321499 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.
%e A321499 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.
%o A321499 (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.
%o A321499 (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.
%o A321499 (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.
%o A321499 (PARI) apply( A321499(n)=if(n<8, 2*n+1, n%5!=3, (n+2)*4\5*2+1, n\5*8+8), [1..30]) \\ Defines A321499(n). The apply() command provides a check & illustration.
%o A321499 (Python)
%o A321499 def A321499(n): return (n<<1)+1 if n<4 else (((n+2)<<2)//5<<1)+(n%5!=3) # _Chai Wah Wu_, Feb 26 2025
%Y A321499 See A321491 for numbers of the form (x+y)(x^2+y^2).
%Y A321499 Cf. A321501 (complement).
%Y A321499 See A321498 for numbers that have two representations of the form (x-y)(x^2-y^2).
%Y A321499 Cf. A106505 (conjectured to be the sequence without the 3).
%K A321499 nonn,easy
%O A321499 1,1
%A A321499 _M. F. Hasler_, Nov 22 2018