A321499 Numbers of the form (x - y)(x^2 - y^2) with x > y > 0.
3, 5, 7, 9, 11, 13, 15, 16, 17, 19, 21, 23, 24, 25, 27, 29, 31, 32, 33, 35, 37, 39, 40, 41, 43, 45, 47, 48, 49, 51, 53, 55, 56, 57, 59, 61, 63, 64, 65, 67, 69, 71, 72, 73, 75, 77, 79, 80, 81, 83, 85, 87, 88, 89, 91, 93, 95, 96, 97, 99, 101, 103, 104, 105, 107, 109, 111, 112, 113, 115
Offset: 1
Examples
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.
Links
- Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).
Crossrefs
Programs
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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(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.
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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.
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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.
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Python
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
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
Comments